zotero/storage/2EHCRV7V/.zotero-ft-cache

Paul Marmet

Einstein's Theory of Relativity

■"

versus

Classical Mechanics

NUNC COCNOSCO EX P4RTE
TRENT UNIVERSITY LIBRARY
PRESENTED BY
PHYSICS DEPARTMENT UNIVERSITY OF OTTAWA

Digitized by the Internet Archive in 2019 with funding from Kahle/Austin Foundation
https://archive.org/details/einsteinstheoryoOOOOmarm

Einstein’s Theory of Relativity versus
Classical Mechanics
By Paul Marmet

Table of Contents

Table of Contents.5

Preface.9

Chapter One

The Physical Reality of Length

Contraction.15 1.1 Introduction.15 1.2 Mass-Energy Conservation at a Macroscopic
Scale.15 1.3 Mass-Energy Conservation at a Microscopic
Scale.17 1.4 Mass Loss of the Electron.20 1.5 Change of the Radius of the Electron Orbit.21 1.6 Change of Energy of Electronic States.22 1.7 Experimental Measurements of Length Dilation
in a Gravitational Potential.24 1.7.1 Pound and Rebka’s Experiment.25 1.7.2 The Solar Red Shift.26
1.8 The Crucial Influence of the Electron Mass on the Fundamental Laws of Relativity.27
1.9 References.28 1.10 Symbols and Variables.29

Chapter
2.1 2.2
2.3 2.4 2.5 2.6 2.7
2.8
2.9

Two Transformation of Excitation Energy
between Frames.30 Introduction..30 Difference between Time and What Clocks Display.30 Description of the Reference Time Rate.32 Description of the Reference Meter.34 Definition of the Velocity of Light.35 Need of Parameters with a Double Index.36 Apparent Lack of Compatibility for Fast Moving Particles.38 Demonstration of the Energy Relationship between Systems.39 Relative Frequencies between Systems.41

6

2.10 Cases of Relevance of the Relationship hv = yhs.43 2.11 Symbols and Variables. 44

Chapter Three Demonstration of the Lorentz Equations

without Einstein's Relativity Principles.. 45

3.1 Fundamental Physical Principle.45 3.2 Change of Energy and Bohr Radius Due to
Kinetic Energy.46 3.3 The Lorentz Equation for Time.48

3.4 Length Dilation Due to Kinetic Energy.52

3.5 The Lorentz Transformation for Lengths.54 3.5.1 Apparent and Absolute Time.55 3.5.2 Relationship between Velocities V and V'.58

3.5.3 Relative Velocities within Systems.58 3.5.4 Lorentz's Second Relationship.61

3.6 Constant Velocity of Light within Any Frame of

Reference.63

3.7 Non-Reality of Space Dilation, Contraction or Distortion.63

3.8 3.9 3.10 3.11

Transformation of Units in Different Frames.65 Failure of the Reciprocity Principle.66 References.68 Symbols and Variables.68

Chapter Four Fundamental Nature of the Mechanism Responsible for the Advance of the Perihelion of Mercury.70
4.1 Definition of the Absolute Standard Units [o.s.].70 4.2 The Absolute Reference Meter.71 4.3 The Absolute Reference Second.74
4.3.1 Example.77 4.3.2 Relative Clock Displays between Frames.78 4.4 The Absolute Reference Kilogram.79 4.5 Space and Time Corollaries within the ActionReaction Principle.80 4.6 Fundamental Mechanism Taking Place in Planetary Orbits.81

7

4.6.1 Significance of Units in an Equation.82 4.7 Transformations of Units.84
4.7.1 aM(o.s.) versus aM(M).84 4.7.2 M(£)(o.s.) and M(M)m(°-s-) versus M(S)(M)
and M(M)m(M).85 4.7.3 PM(o.s.) versus Pm(M).87 4.7.4 G(o.s.) versus G(M).88 4.7.5 F(o.s.) versus F(M).90 4.8 Symbols and Variables.90

Chapter Five Calculation of the Advance of the Perihelion

of Mercury.93

5.1
5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13

Mathematical Transformation of Units between Frames.93
5.1.1 Consequence of a Simple Change of Units.94 Physical Transformations Due to Mass-Energy Conservation.94 Incoherence between Outer Space and Mercury Predictions Using Newton's Physics..96 Incoherence of the Gravitational Force Using Newton's Physics.97 Relevant Physical Parameters.99 Fundamental Phenomena Responsible for the Advance of the Perihelion of Mercury.100 Change of Length from Outer Space to Mercury Location.101 Change of Clock Rate from Outer Space to Mercury Location.104 Total Interaction Due to the Physical Changes of Length and Clock Rate.106 Correction for an Elliptical Orbit.108 Mathematical Identity with Einstein’s Equation.110 References.Ill Symbols and Variables.Ill

8
Chapter Six Geometrical Illustration of the Advance of the Perihelion of Mercury.113
6.1 Conditions Controlling the Geometrical Shape of an Orbit.113
6.2 The Change of Mass of Mercury.115 6.3 Orbital Shapes and Gravitational Force Gradients ..117 6.4 Identity of Mathematical Forms.119 6.5 Illustration of Trajectories in Potential Wells.119 6.6 Validity of the Classical Model.121 6.7 References.122 6.8 Symbols and Variables.122
Chapter Seven The Lorentz Transformations in Three Dimensions.124
7.1 Basic Principles of a Transformation.124 7.2 The Lorentz Transformations.127 7.3 The Equations.127 7.4 Symbols and Variables.128
Chapter Eight The Doppler Effect.129 8.1 Fundamental Principles of the Doppler Effect.129 8.2 Mass-Energy Conservation in the Context of the Doppler Effect.130 8.3 The Doppler Effect without Using Waves.130 8.4 References.132 8.5 Symbols and Variables.133
Chapter Nine Simultaneity and Absolute Velocity of Light.134
9.1 Simultaneity versus Identical Clock Displays.134 9.2 Thought Experiment on Clocks Synchronization.... 134 9.3 Synchronization of Clocks A and B.135
9.3.1 Method #1.135 9.3.2 Method #2.135 9.4 Loss of Synchronization of Clock a on the Moving Frame.137

9

9.5 Synchronization between Moving Clocks a and p (Method #1).138
9.6 Asymmetric Relative Velocity of Light.140 9.7 Synchronization of Clocks a and p (Method #2).142 9.8 References.145 9.9 Symbols and Variables.145

Chapter Ten

The Principle of Equivalence.146

10.1 Introduction.146

10.2 Deflection of Light in an Elevator Moving at

Constant Velocity.146

10.3 Inertial versus Gravitational Acceleration of

Masses.147

10.4 Bremsstrahlung Due to Inertial and Gravitational

Accelerations.149

10.5 Behavior of Light.151

10.5.1 Light Path in an Accelerated Elevator.151

10.5.2 Light Path in a Gravitational Field.152

10.5.3 The Equivalence Principle and Light

Deflection.154

10.6 Gravitational Lenses.155

10.7 Attracting Force between Parallel Beams of

Charged Particles.156

10.8 References.157

Chapter Eleven Internal Phenomena inside Atoms.158 11.1 Introduction..158 11.2 Transformations inside Fast Moving Atoms.158 11.3 Electric Potentials.159 11.4 Sommerfeld Fine Structure.162 11.5 Atomic Structure inside Free Falling Atoms.165 11.6 High Potentials and Higher Order Terms.166 11.7 References.167 11.8 Symbols and Variables.167

Chapter Twelve On the Formation of Pseudo Black Holes.168
12.1 Formation of a Protostar.168

10

12.2 Mass-Energy Conservation in a Cluster of Atoms... 169 12.3 Mass of a Star versus the Amount of Matter Used
for Its Formation. 173 12.4 Mass of a Star versus Its Radius.174 12.5 Maximum Mass of a Star versus Its Radius.176 12.6 Complete Transformation of Mass into Energy.176 12.7 Proper Values in Extreme Gravitational
Potentials.178 12.8 Beyond the Extreme Gravitational Potential.179 12.9 Formation of Matter in a Deep Gravitational
Potential versus the Formation of Matter and Anti-Matter.180
12.9.1 Inverse Gravitational Mechanism.181
12.10 References.182

Appendix I

The Dependence of the Size of

Matter on Electron Mass.184

Appendix II The Deflection of Light by the Sun's

Gravitational Field: An Analysis of

the 1919 Solar Eclipse Expeditions.189

Appendix III Physical Constants.197

Index.198

PREFACE
The aim of this book is to demonstrate that using "Conventional Wisdom" and "Conventional Logic", classical physics can explain all the observed phenomena attributed to relativity. The arbitrary principles of Einstein's relativity are thus useless.
It is very important to recognize the fundamental importance of the principle of mass-energy conservation. It took thousands of years of development for scientific thought to finally reject the magic of witchcraft. During the nineteenth century, scientists became convinced that matter cannot be created from nothing. Conversely, matter cannot be destroyed into nothing. It seems that even Einstein believed this, since he is the one who, at the beginning of the twentieth century, introduced the equation E = me implying mass-energy conservation. However, he later developed general relativity which is not compatible with that principle. Indeed, according to Straumann1, the:
"general conservation law of energy and momentum does not exist in general relativity".
Twentieth century science moved backward in accepting again the magical creation of matter or energy from nothing, even if this is hidden in complicated mathematics.
Contrary to what Einstein did, all the demonstrations in this book are compatible with the principle of mass-energy and momentum conservation. Using classical mechanics, we demonstrate that length contraction is a real physical phenomenon. We examine how this leads to the Lorentz equations. Then, we show how classical principles are sufficient to explain the advance of the perihelion of Mercury and derive Einstein's equation. The fundamental reason for this advance is illustrated with a classical
1 Straumann, N., General Relativity and Relativistic Astrophysics. Springer-Verlag, Berlin, 1991, page 146.

12
apparatus. We also study the Lorentz transformations in three dimensions and the Doppler phenomenon. Then we see how the problems brought by the relativity of simultaneity and by the principle of equivalence can be explained using conventional logic. We also show how classical mechanisms produce perturbations in the internal structure of atoms and molecules. Finally, we show that the presence of intense gravitational potentials leads to degenerate matter corresponding to Schwarzschild's black holes.
Einstein's relativity principles are not needed in these demonstrations. The only principles used are the ones already existing in classical mechanics. All the solutions are based on a physical model compatible with conventional logic.
Einstein's theory of relativity is a mathematical model which is not compatible with the physical models described in classical mechanics since it is not compatible with the principle of massenergy conservation. This is a well-known fact. It is claimed that the theory of relativity is so advanced that it is not possible to give a Newtonian physical description of it. It is also often argued that abandoning classical scientific concepts leads to a scientific revolution. It is erroneous to believe that a new scientific revolution must abandon the fundamental principles brought up by Newton's classical mechanics and logic which gave birth to all our knowledge in physics.
As stated in several papers, Einstein's relativity implies "New Logic" which contradicts "Conventional Logic". Einstein's theory implies that because we can find some arbitrary mathematical relationships that fit some experiments, we must abandon conventional logic. History reports some rudimentary scientific models that also fitted experiments but which were based on nonsense. Those models were rejected. A new scientific revolution based on "New Non Conventional Logic" can lead to a scientific disaster or to a dead end. No scientific concept can be so advanced that it is no longer compatible with logic.
Einstein's relativity assumes new mathematical hypotheses and ignores completely the concept of models to describe physical reality. Einstein supposed that time and space can be distorted and that simultaneity is relative but he did not give any serious

13
description of what this really means physically. In Newton's time, physical descriptions of phenomena were accompanied by mathematical equations giving quantitative predictions corresponding to those physical descriptions. Einstein's relativity claims that nature can be described with mathematical equations without any physical description. There is a complete abandon of all the physical models that made physics understandable in Newton's time.
Our main argument here is not whether Einstein's hypotheses are true or not. We believe that if Einstein's hypotheses are correct, they must correspond to a real physical mechanism. Such a real mechanism is described in this book using classical mechanics and classical logic.
With Einstein's new logic, contradictory results have appeared. For example, Gerald Feinberg developed the theory of tachyons which move faster than the speed of light. There are also mathematical models calculating wormholes, strings, multidimensional space, superluminal objects, time reversal and even time lines. Certainly, these claims do not make sense when we use conventional logic.
An expert in Einstein's relativity is described as an expert in the mathematics of relativity. Since the conventional wisdom of classical physics is not used in relativity, an expert in relativity is not trained to deal with Newtonian logic. Consequently, this book on relativity will be much more easily understood by an expert in classical physics since he or she already knows the mathematics and understands the classical mechanisms involved. It might appear surprising to some readers that relativity can be explained with classical principles. However, they will never escape out of their preconceived notions and learn how this is done unless they carefully read this book.
Acknowledgments.
The author wishes to express his gratitude to Christine Couture tor writing the appendixes and for her skill in preparing the illustrations. As a physicist, she initiated many successful discussions regarding the content and the editorial work in this

14
book. Some philosophical discussions with J.-C. Gille and A. StJacques were indispensable to develop the basic ideas leading to physical reality. The author is also grateful to Bruce Richardson for his interest in fundamental science and his financial support at a critical time. Collaboration was much appreciated from Drs Y. Varshni, M. LeBlanc and B. Hird. Various help related to programming, computer work or precious encouragements were received from Nancy Robertson, Nicolas Marmet, G. Y. Dufour and most importantly from my wife Jacqueline.
/

Chapter One The Physical Reality of Length Contraction.
1.1 - Introduction.
In this first chapter, we will show that it is possible to establish links between quantum mechanics and mass-energy conservation. These links will help us calculate the interatomic distances in molecules and in crystals as a function of their gravitational potential. We will show that the natural interatomic distance calculated using quantum mechanics leads to the length contraction (or dilation) predicted by relativity. This result will be obtained here without using the hypothesis of the constancy of the velocity of light. It will appear instead as a consequence of quantum mechanics when mass-energy conservation is taken into account.
Since length contraction appears as a consequence of quantum mechanical calculations, the physical reality of those predictions can be verified experimentally. We will show that the results of the most precise quantum mechanical experiments prove that the change of length is real. Two different experiments which have been found to give sufficient accuracy to verify this change of length will be described in detail. We will show that the dimensions of matter really change naturally depending on its location in a gravitational potential.
1.2 - Mass-Energy Conservation at Macroscopic Scale.
The most reliable principle in physics seems to be the principle of mass-energy conservation: mass can be transformed into energy and vice versa. Without this principle, one would be able to create mass or energy from nothing. We do not believe that absolute creation is possible.
In order to understand the fundamental implications related to mass-energy conservation, let us consider the following example. Suppose momentarily that the Earth is not moving around the Sun, but has been pushed away with a powerful rocket and has reached

16

CHAPTER ONE. Reality of Length Contraction.

interstellar space at location P (see figure 1.1). It now has a negligible residual velocity with respect to the Sun and except for the fact that the Sun has faded away, everything appears the same. The Earth is still made of about 1050 atoms, its center contains iron, is surrounded by oceans, deserts, cities and the atmosphere is the same. The planet is still populated by about the same five billion people.

Figure 1.1
Let us assume that after a while, the planet starts falling slowly from P toward the Sun. Due to the solar attraction, the Earth accelerates until it reaches the distance of 150 million kilometers (from the Sun) corresponding to its normal orbit. At that moment, one can calculate that the Earth has reached a velocity of 42 km/s. This velocity is too large for the Earth to be in a stable orbit around the Sun as it is normally. It must be reduced to 30 km/s, the velocity for a stable orbit. The Earth must be slowed down.
It is decided that the velocity of the Earth can be reduced with the help of a strong rope attached to a group of stars at the center of our galaxy. The force produced by the rope will generate energy at the center of the galaxy while the Earth is slowed down to the desired velocity for a stable orbit around the Sun.
Knowing that the Earth has a mass of 5.97x10"4 kg, it is easy to calculate the amount of work transferred to the center of the galaxy. It corresponds to slowing down the Earth from 42 km/s to 30 km/s. This represents an amount of work equal to 2.6x1033 joules. Therefore the Earth must get rid of 2.6x1033 joules to go back to its normal orbit and the center of the galaxy must absorb that same amount of energy. The rope used to slow down the Earth could then run a generator located at the center of the galaxy to produce 2.6x1033 joules of energy.

CHAPTER ONE. Reality of Length Contraction.

17

However, due to the principle of mass-energy conservation, the energy carried out to the center of the galaxy to slow down the Earth can be transformed into mass. Using the relation E = me2, we find that the mass corresponding to 2.6x 1033 joules of energy is equal to 2.9x1016 kg. This means that 29 billions of millions of kilograms of mass have been transferred from the Earth to the center of the galaxy through the rope. This mass-energy is a very small fraction of the Earth’s mass but it must be coming from the Earth and received at the center of the galaxy.
After the re-establishment of the Earth’s orbit at one astronomical unit from the Sun, the inhabitants of the Earth find nothing changed. Other than the neighboring Sun, no difference can be noticed compared with when the Earth, still made of its initial 1050 atoms, was away from the Sun. The question is: How can the Earth not lose one single atom or molecule while 29 billions of millions of kilograms of mass have been lost and received at the center of the galaxy? There is only one logical answer. Since each atom on Earth was submitted to the force of the rope, each atom has lost mass in a proportion of approximately one part per one hundred million.
Note that this situation is equivalent to the formation of a hydrogen atom. When a proton and an electron come together to form a hydrogen atom, energy is released in the form of light. This light corresponds to the work transferred to the center of the galaxy in our problem.
1.3 - Mass-Energy Conservation at a Microscopic Scale.
The experiment described above takes place at a macroscopic scale. Each individual atom loses mass because a force interacts on all atoms when the Earth decelerates in the Sun's gravitational potential. It is normally assumed that atoms have a constant mass. For example we learn that the mass of the hydrogen atom is m0 = 1.6727406xl0"27 kg. Can we have hydrogen atoms with less or more mass? From the thought experiment of section 1.2, we see that the principle of mass-energy conservation requires a transformation of mass into energy on each atom forming the

18

CHAPTER ONE. Reality of Length Contraction.

Earth, since each of them has contributed to generate energy transmitted to the center of the galaxy.

Let us study the following experiment. We first consider that an individual hydrogen atom is placed on a table on the first floor of a house in the gravitational field of the Earth, as shown on figure 1.2. The hydrogen atom is then attached to a fine (weightless) thread so that the atom can be lowered down slowly to the basement of the house, while the experimenter remains on the first floor. When the atom is lowered down, its weight produces a force F in the thread. That force is measured by the experimenter on the first floor. It is given by:

F = m0g.

1.1

First floor m„

Ah Basement

Figure 1.2

The slow descent of the atom attached to the thread is stopped every time a measurement is made, which means that the kinetic energy is zero at the moment of the measurement. When the atom has traveled a vertical distance Ah, the observer on the first floor observes that the energy AE produced by the atom and transmitted through the thread to the first floor is:

AE = FAh.

1.2

The work extracted from the descent of the atom is positive when the final position of the atom is under the first floor (Ah is positive). Then, according to the principle of mass-energy conservation, the energy produced at the first floor by the descent of the atom in the basement can be transformed into mass according to the relationship:

CHAPTER ONE. Reality of Length Contraction.

19

bc = me2.

1.3

The important point that must be retained about equation 1.3 is that the energy E is proportional to the mass, independently of the fact that it just happens that the numerical value of the constant of proportionality is equal to the square of the velocity of light. From equations 1.1, 1.2 and 1.3, the amount of mass Amf generated at the first floor by the descent is:

Amf =

mogAh

1.4

This amount of mass (or energy) carried by the thread is generated by the weight of the atom which slowly moves down to the basement. When the hydrogen atom lies on the table, its mass is m0. However, during its descent, it produces work (corresponding to the mass Amf generated at the first floor). The initial mass m0 of the particle is now transferred into the massenergy Amf generated at the first floor by the falling particle, plus the remaining mass mb of the particle now in the basement. Using equation 1.4, we find:

mb = m0 - Amf = m0

1.5

According to the principle of mass-energy conservation, the mass of the hydrogen atom in the basement is now different from its initial mass m0 on the first floor. It is slightly smaller than mQ and is now equal to mb. Any variation of g with height is negligible and can be taken (with g) into account in equations 1.4 and 1.5.
lb
Of course, the relative change of mass Am/m0 is extremely small. (It was equally small in the case of the Earth falling back to its normal orbit, as seen above in section 1.2.) The change of mass given by equation 1.5 is so small that it cannot be verified using a weighing scale. However, this reduction of mass must exist, otherwise, mass-energy would be created from nothing. We will see below that this change of mass has actually been measured.
It was quite arbitrary for us to assume that the initial mass of hydrogen on the first floor is m0. Physical tables do not mention all the experimental conditions in which an atom is measured.

20

CHAPTER ONE. Reality of Length Contraction.

Furthermore, the accuracy of this value is quite insufficient now to detect Amf (equation 1.5). A change of altitude of one meter near the Earth’s surface gives a relative change of mass of the order of lO'16. Masses are not known with such an accuracy.
At this point, we must recall that in the above reasoning, we have made a choice between the principle of mass-energy conservation and the concept of absolute identical mass in all frames. It is illogical to accept both principles simultaneously since they are not compatible. We have chosen to rely on the principle of mass-energy conservation which is equivalent to not believing in "absolute creation from nothing" as defined in section 1.2. We must realize that without mass-energy conservation not much of physics remains. Physics becomes magic.

1.4 - Mass Loss of the Electron.
There is a way to measure experimentally the mass difference between a hydrogen atom in the basement and one on the first floor. In equation 1.5, we see that a mass Amf appears and increases when the atom moves down in the gravitational field. Due to mass-energy conservation, the mass mb of the atom moving down decreases by the same amount, that is:

Amb = Amf

1.6

Since the hydrogen atom has lost a part of its mass due to the change of gravitational potential energy, we must expect (according to equation 1.5) that the electron as well as the proton in the atom have individually lost the same relative mass. Let us calculate the relative change of mass of the electron fAmg/me) and of the proton inside the hydrogen atom due to its change of height.
From equations 1.5 and 1.6, we have:

Ame gAh

-= —

1.7

me c2

where

Ame = Am,,.

1.8

CHAPTER ONE. Reality of Length Contraction .

21

When Ah is a few meters, equation 1.7 gives a relative change of mass of the order of 10’16. Consequently, the first order term gives an excellent approximation. Let us use:
Ame dmc

The electron mass me (as well as the proton mass) is not constant and decreases continuously when the atom is moving down. Equation 1.7 shows that independently of the mass of the particle, the relative change of mass is the same. This means that for the same change of altitude, the relative change of mass of the electron is the same as for the proton.
Due to the principle of mass-energy conservation, we must conclude that a hydrogen atom at rest has a less massive electron and a less massive proton at a lower altitude than at a higher altitude. The mass of an electron and of a proton can be tested very accurately in atomic physics. Quantum physics shows us how to calculate the exact structure of the hydrogen atom as a function of the electron and proton mass. From that, one can calculate the Bohr radius of an atom having a different mass. Fortunately, the Bohr radius can also be measured with extreme accuracy experimentally.
1.5 - Change of the Radius of the Electron Orbit.
It is shown in textbooks how quantum physics predicts the radius of the orbit of the electron in hydrogen for a given electronic state. This is given by the well known Bohr equation:
n2h2 1.10
n Zmeke2
where rn is the radius of the Bohr orbit of the electron with principal quantum number n, me is the mass of the electron (actually, me is the reduced mass, but it is approximately the same as the electron mass), h is the Planck constant (= 2ti/z), k is the Coulomb constant (l/47te0), e is the electronic charge and Z is the number of charges in the nucleus (Z = 1 corresponds to atomic hydrogen). Furthermore when we choose n = 1 and Z = 1, rn becomes a0, which is called the Bohr radius. The Bohr radius is

22

CHAPTER ONE. Reality of Length Contraction.

5.291772x10'" m at the Earth's surface (for the case of for which the nucleus is very massive). Equation 1.10 illustrates a simple principle. It illustrates the fact that the circumference of the electron orbit is exactly equal to (or any multiple of) the de Broglie wavelength of the electron orbiting the nucleus.
Since, as we have seen above, the electron mass me changes with its position in a gravitational potential, let us calculate (using Bohr's equation) the change of radius rn caused by that change of electron mass. This is given by the partial derivative of rn with respect to me. From equation 1.10 we find:

<K = __^e

i n

rn

me

Equation 1.11 shows that any relative decrease of electron mass is equal to the same relative increase of the radius of the electron orbit. According to the principle of mass-energy conservation, the

electron mass decreases when brought to a lower gravitational

potential. Consequently, quantum physics (Bohr's equation) shows that the radius of the electron orbit in hydrogen must increase

when the atom is at a lower altitude. Using equation 1.10,

quantum physics gives us the possibility to predict the size of the

electron orbit rn in an atom for different values of electron mass. Let us study the change of size of the electron orbit as a function of the altitude where the particle is located in a gravitational field.

1.6 - Change of Energy of Electronic States.
Since it has been observed and accepted that the laws of quantum physics are invariant in any frame of reference, let us calculate the energy states of atoms having an electron (and a proton) with a different mass. The consequences of the change of proton mass are easily calculated since the energy levels depend only on the reduced mass of the electron-proton system. In the Bohr equation, we take me as the reduced mass. This does not produce any relevant difference in the problem here.
The binding energy between the electron and the proton is a function of the electrostatic potential between the nucleus and the electron. Quantum physics teaches that the energy En of the nth state as a function of the electron mass is:

CHAPTER ONE. Reality of Length Contraction.

23

f k2e4 ^

En = 2n 2 /+i 2 m e‘

1.12

From equation 1.12, we can find the relationship between the change of electron mass and the change of energy:

5m„

En

me

1.13

The Bohr radius a0 is the average radius of the electron orbit for n = 1. According to quantum physics the energy of state n is:

1.14

where a0 is a function of the electron mass me, given by:

K1 = Ike2 Jvf-mLjl

1.15

We know that the energy of electronic states of atoms can be measured very accurately in spectroscopy from the light emitted during the transition between any two states En and En.. Extremely accurate results can also be obtained in some nuclear reactions with the help of Mossbauer spectroscopy.

The frequency vn of the radiation emitted as a function of the energy En of level n is given by:

En = hvn.

1.16

By differentiation of equation 1.16, we find:
dv„ 3En 1.17
W En
Differentiation of equation 1.14 gives:
3Er da„ 1.18

Combining equations 1.11, 1.13, 1.17 and 1.18, we get:

da,

Svr

<3En drn

mP

1.19

24

CHAPTER ONE. Reality of Length Contraction.

Since these quantities are extremely small but finite, we can

write:

A<x,

Av„

Am,

AEn Arn

1.20

a0

vn

me

From equation 1.7, we have:

Ame gAh 1.21
me c‘

Equations 1.20 and 1.21 give:

AEn Avn En vn

Aa0 Ame a0 me

Arn gAh

rn

c2 •

1.22

Equation 1.22 shows that the relative change of size of the Bohr radius Aa0/a0 is equal to -gAh/c2.

This shows that following the laws of quantum physics, a change of electron mass due to a change of gravitational potential (which results necessarily from the principle of mass-energy conservation) produces a physical change of the Bohr radius.

We must notice here that using the relativistic correction given by Dirac's mathematics is irrelevant and does not solve this problem. Relativistic quantum mechanics introduces a relativistic correction due to the electron velocity with respect to the center of mass of the atom. The change in electron mass brought by the relativistic correction involved in this chapter is due to the gravitational potential originating from outside the proton-electron system. It is not due to any internal velocity within the atom. The use of the relativistic Dirac equation is not related to calculating how the Bohr radius changes between its value in the initial gravitational potential and its value in the final gravitational potential.

1.7 - Experimental Measurements of Length Dilation in a Gravitational Potential.
A measurement proving that there is a change of the Bohr radius due to the change of gravitational potential has already been made. The difference of energy for an atom corresponding to its change

CHAPTER ONE. Reality of Length Contraction.

25

of size is observed as a red shift of its spectroscopic lines. The change of mass can be applied quite generally to any particle or subatomic particle in physics placed in a gravitational potential. It can also be applied to astronomical bodies like planets and galaxies since it relies on the principle of mass-energy conservation which is always valid.

1.7.1 - Pound and Rebka’s Experiment.
A spectroscopic measurement of the highest precision has been reported by Pound and Rebka [1] in 1964 with an improved result by Pound and Snider in 1965. Since we have seen that the change of a0 corresponds to a change of energy of spectroscopic levels, let us examine Pound and Rebka's experiment. They used Mossbauer spectroscopy to measure the red shift of 14.4 keV gamma rays from Fe . The emitter and the absorber were placed at rest at the bottom and top of a tower of 22.5 meters at Harvard University.

The consequence of the gravitational potential on the particles is such that their mass is lower at the bottom than at the top of the tower. Therefore an electron in an atom located at the base of the tower has a larger Bohr radius than an electron located 22.5 meters above, as given by equation 1.22. The same equation also shows that electrons orbiting with a larger radius have less energy and emit photons with longer wavelengths.

Pound and Rebka reported that the measured red shift agrees within one percent with the equation:

AE

:^ = 2.5x1 O'15.

1.23

E

Not only is the change of energy predicted by relativity and verified experimentally by Pound and Rebka (equation 1.23)

numerically compatible with the change of energy predicted by the

conservation of mass-energy, but the predicted relativistic equation

is mathematically identical to the one predicting the increase of

Bohr's radius (equation 1.22). Since the red shift measured corresponds exactly to the change of the Bohr radius existing

between the source and the detector, we see that it cannot be

attributed to an absolute decrease of energy of the photon during

its trip in the gravitational field.

26

CHAPTER ONE. Reality of Length Contraction.

This result is exactly the one that proves that matter at the base of the tower is dilated with respect to matter at the top. It is clear that the Bohr radius has actually changed as expected which means that the physical length has really changed. Therefore, this phenomenon is not space dilation. The real physical dilation of matter is observed because electrons (as well as all particles) have a lower mass at the bottom of the tower which gives them a longer de Broglie wavelength. Space dilation is not compatible with a rational interpretation of modem physics. A rational interpretation has already been presented [3].
The equilibrium distance between particles is now increased because the Bohr radius has increased. When atoms are brought to a different gravitational potential, the electron and proton must reach a new distance equilibrium as required by quantum physics in equation 1.12. Quantum physics and the principle of massenergy conservation lead to a real physical contraction or dilation. This solution solves the mysterious description of space contraction in relativity without involving any new hypothesis or new logic. Length contraction or dilation is real and is demonstrated here as the result of actual experiments. Let us also note that this length dilation is done without producing any internal mechanical stress in solid material. Finally, if the source were above the detector, we would observe a blue shift proving that the Bohr radius in matter above the detector has decreased with respect to the Bohr radius in matter at lower altitude. One can conclude that Pound and Rebka's experiment has shown that matter is contracted or dilated when it is moved to a different gravitational potential.
1.7.2 - The Solar Red Shift.
Other experiments also show the reality of length contraction or dilation. For example, the atoms at the surface of the Sun have been measured to show exactly the gravitational dilation due to the decrease of mass of the electrons in the solar gravitational potential. The gravitational potential at the Sun's surface is well known. As shown above, it is a change of electron mass in the hydrogen atom due to the gravitational potential that produces a change of the Bohr radius. It is that change of Bohr's radius that

CHAPTER ONE. Reality of Length Contraction.

27

produces a change of energy between different atomic states. Brault [2] has reported such a change of energy between atomic states. It corresponds exactly to the change of Bohr's radius caused by the gravitational potential. The atoms on the Sun emit light at a different frequency because the electrons are lighter on the solar surface than on Earth, exactly as required by the principle of massenergy conservation. The change of electron mass on the Sun produces displaced spectral lines toward longer wavelengths as given by equation 1.22. Since quantum physics is valid on the solar surface, we can understand that the electrons have less mass due to the solar gravitational potential. This leads to an increase of the Bohr radius for the atoms located on the solar surface which leads to atomic transitions having less energy, as observed experimentally.
The Mossbauer experiment as well as the solar red shift experiment prove that atoms are really dilated physically. This means that the physical length of objects actually changes. We also find that not only do protons and electrons lose mass in a gravitational potential but so do nuclear particles in the nucleus of Fe57, as observed in the Mossbauer experiment of Pound and Rebka.
1.8 - The Crucial Influence of the Electron Mass on the Fundamental Laws of Relativity.
Macroscopic matter is formed by an arrangement of atoms. In molecular physics, we learn that quantum physics predicts that interatomic distances are proportional to the Bohr radius. Those distances are calculated as a function of the Bohr radius. According to quantum physics, a smaller Bohr radius will lead to a smaller interatomic distance between atoms in molecular hydrogen. The interatomic distance in molecules is known to be a function of the Bohr radius just as the interatomic distance in a crystalline structure is proportional to the Bohr radius. This means that since the Bohr radius changes with the intensity of the gravitational potential, the size of molecules and crystals also changes in the same proportion. This is true even in the case of large organic molecules. Therefore the size of all biological matter

28

CHAPTER ONE. Reality of Length Contraction.

is proportional to the Bohr radius. This point is explained in more details in appendix I.
Because the size of macroscopic matter changes with the gravitational potential, the original length of the standard meter transferred to a location having a different gravitational potential will also change. To be more specific, mass-energy conservation requires that the standard meter made of platinum-iridium alloy becomes shorter if we move it to the top of a mountain. Furthermore, due to the increase of electron mass, an atomic clock will increase its frequency by the same ratio when it is moved to the top of the same mountain. However, since the velocity of light (or any other velocity) is the ratio between these two units, it will not change at the top of the mountain with respect to any frame of reference. This point will be discussed later. Because the relative changes of length and clock rate are equal, they will be undetectable when simply using proper values within a frame of reference. All matter, including human bodies, composed of atoms and molecules will change in the same proportion since the intermolecular distance depends on the Bohr radius and consequently on the electron mass which is reduced when located in a gravitational potential.
It is important to notice that length dilation or contraction is predicted and explained here without using the relativistic Lorentz equations nor the constancy of the velocity of light. Consequently, we must consider now that we have demonstrated experimentally (using Pound and Rebka's results) the physical change of length of an object in a gravitational potential. More demonstrations will be given in the following chapters.
The experiments reported here showing length dilation use atoms that are at rest. They are solely related to the potential energy. We will see that the problems of kinetic energy and velocities require new considerations in the next chapters.
1.9 - References.
[1] C. W. Misner, K. S. Thome and J. A. Wheeler, Gravitation. W. H. Freeman and Company San Francisco, page 1056. See also: Pound R. V. and G. A. Rebka, Apparent Weight of Photons. Phys.

CHAPTER ONE. Reality of Length Contraction.

29

Rev. Lett., 4, 337 1964. See also: Pound R. V. and Snider, J.L. Effect of gravity on Nuclear Resonance. Phys. Rev. B, 140, 788-
803, 1965. This has been measured in a rocket experiment by Versot and Levine (1976) with an accuracy of 2 x 10'4.
[2] J. W. Brault, The Gravitational Redshift in the Solar Spectrum. Doctoral dissertation, Princeton University, 1962. Also Gravitational Redshift in Solar Lines. Bull. Amer. Phys. Soc., 8, 28, 1963.
[3] P. Marmet, Absurdities in Modem Physics: A Solution. ISBN 0-921272-15-4 Les Editions du Nordir, c/o R. Yergeau, 165 Waller, Ottawa, Ontario KIN 6N5, 144p. 1993.

1.10 - Symbols and Variables.
AE energy produced by the atom and transmitted to the first floor
Ah distance travelled by the atom Amb amount of mass lost by the atom Ame amount of mass lost by the electron Amf amount of mass generated on the first floor En energy of the hydrogen atom in state n F weight of the atom m0 mass of the atom on the table vn frequency of the radiation emitted corresponding to En rn radius of the orbit of the electron in hydrogen in state n Z number of charges in the nucleus

Chapter Two Transformation of Excitation Energy between
Frames.

2.1 - Introduction.
We consider now the kinetic energy given to masses when there is no gravitational potential. The principle of mass-energy conservation requires that masses increase when given kinetic energy. This is expressed by the relationship:

mv[rest] = yms[rest]

2.1

where:

V|2
2.2 vcy

The index [rest] means that the measurement is made using the units of the rest frame. The subscripts v and s refer to masses having respectively a velocity v and no velocity (stationary). These indices will be explained in detail in section 2.6.

Since masses can be excited particles containing internal potential energy, we must study how to transform that potential energy between frames. The mass-equivalent of this internal potential energy has always been ignored in relativity. In order to be coherent, it must be taken into account. Let us show how this correction restores physical reality in relativity. To calculate the relationship between masses in different frames we use the principle of mass-energy conservation (equation 2.1). Let us find an equivalent relationship for the case of energy released by an excited atom.

2.2 - Difference between Time and What Clocks Display.
It has been suggested that time is what clocks measure. This definition is incomplete and misleading. We have seen in chapter one that due to mass-energy conservation, clocks in different gravitational potentials run at different rates. We must realize that

CHAPTER TWO. Transformation of Excitation Energy between Frames. 31
"time" is not elapsed more slowly because a clock functions at a slower rate or because the atoms and molecules in our body function at a slower rate.
We have seen in equation 1.22 that in the case of a change of gravitational potential, the Bohr radius is larger when the electron mass is smaller. We also know that according to quantum mechanics, atomic clocks run more slowly when the electron mass is smaller. When we say that an atomic clock runs more slowly, we mean that for that atomic clock, it takes more "time" to complete one full cycle than for an atomic clock in the initial frame, where the electron has a larger mass. That slower rate can only be measured by comparing the duration of a cycle in the initial frame with the duration of a cycle in the new frame. It is the time rate measured in the initial frame at rest that is considered the "reference time rate". We will see that all observations are compatible with this unchanging "reference time rate".
The change of clock rate is not unique to atomic clocks. We recall that quantum mechanics shows that the intermolecular distances in molecules and in crystals are proportional to the Bohr radius (see appendix I). Consequently, due to velocity, the length of a mechanical pendulum will change. Therefore it can be shown that the period of oscillation of all clocks (electronic or mechanical) will also change with velocity.
We cannot say that "time" flows at the rate at which all clocks run because not all clocks run at the same rate. However, a coherent measure of time must always refer to the reference rate. That reference rate corresponds to the one given by a reference clock for which all conditions are fully described. It never changes. However, all matter around us (including our own body) is influenced by a change of electron mass (see appendix I) so that we are deeply tied to the rate of clocks running in our frame. Since our body and all experiments in our frame are closely synchronized with local clocks, it is much more convenient to describe the results of experiments as a function of the clock rate in our own frame. This is what we call the "apparent time".
We generally refer to the clock rate of our organism believing that we are referring to the "real time". What appears as a "time

32 CHAPTER TWO. Transformation of Excitation Energy between Frames.
interval" for our organism is in fact the difference between two "clock displays" on a clock located in our own frame. "Difference of clock displays" (ACD) is a heavier phrase than "time interval" but it is necessary for an accurate description of nature. Of course, clocks are instruments measuring time but during the same time interval there is a difference by a factor of proportionality between the "differences of clock displays" of different frames. In order to avoid any misinterpretation, we must use the word "time" with great caution when we want to shorten the description. In that case, "time" is an apparent time interval corresponding to the difference of clock displays in a given frame when no correction has been made to compare it with the reference time. Since all our clocks and biological mechanisms depend on the electron's mass and energy, humans feel nothing unusual when going to a new frame. However, the time measured by the observer in that new frame is an apparent time and it must be corrected to be compared with a time interval on the fundamental reference frame.
2.3 - Description of the Reference Time Rate.
We do not know how to build a clock whose rate will not change when brought to a different gravitational potential or to a different velocity. However, using the mass-energy conservation principle, we have seen in equation 1.22 how to calculate the difference of clock rate between clocks without relative velocity and located in different gravitational potentials. This means that we can calculate the clock rate in one frame as a function of the clock rate in a different frame, as long as the gravitational potential and kinetic energies are fully described in both frames.
An absolute "reference time rate" can be defined using a clock located in a frame in which the velocity and the gravitational potential are well described. For example this could be a clock at rest with respect to the Sun and far enough from it so that the residual gravitational potential would be negligible. We could then arbitrarily define the "reference time rate" as the rate at which that clock operates in these particular conditions. Everywhere in the universe we would refer to that rate as the "reference time rate". If such a reference clock were brought from outer space to a location near the Sun, we have found in chapter one that due to mass-energy

CHAPTER TWO. TRANSFORMATION OF EXCITATION ENERGY BETWEEN FRAMES. 33
conservation, it would run more slowly because the electron would lose mass into energy that would escape away from its initial frame.
Let us assume that an observer near the Sun wants to measure the period of variation of light coming from a remote variable star. He uses his clock and records a clock display every time the star is at its maximum of brightness. The difference between two maxima will give him the period of variation of the star, using his clock rate. Let us represent by ACDS (where s stands for Sun) the difference of clock displays for the clock near the Sun. In Einstein's relativity, since time is what clocks measure, ACDS is interpreted as a time interval. However, we know that a difference of clock displays simply gives a pure number without any information on what the absolute time is. The subscript of ACDS refers only to the location of the clock and not to an absolute time unit. We know however that another clock far away from the Sun (in a higher gravitational potential) will give a different difference of clock displays called ACD0 s (where o.s. stands for outer space) between each maxima because it runs at a different rate (that is equal to the "reference outer space clock rate"). Consequently, the ACDS recorded near the Sun will not be the same as the ACD0 s recorded in outer space. The observer near the Sun will have the illusion of a "time interval" (that he might call At) that is different from the one measured by the observer located in outer space simply because the clock rate at his location is different due to a different electron mass. One must understand that the real time interval for a star to complete a cycle does not vary because the observer has moved somewhere else or because his clock runs at a different rate. Consequently, when we refer to ACD, we must always specify (with a subscript) in which frame the clock is located. Then a correction needs to be made to that number if we want to calculate the corresponding ACD given by a reference clock in outer space. We must remember that the ACD given by a local clock is a pure number that must be multiplied by a unit of time to give a "real time" interval. Therefore, an absolute reference of "time unit" must be defined. Furthermore, the absolute standard of unit of time will appear different in different frames since we

34 CHAPTER TWO. Transformation of Excitation Energy between Frames.
have seen that local clocks run at different rates in different gravitational potentials.
We see that there is no time dilation nor time contraction. There is no magic. In order to be able to make a comparison between systems, it is absolutely necessary to compare the differences of clock displays (which are not time but numbers of units of time) instead of the time intervals.
This problem cannot be discussed properly using directly the parameter "time" because of the psychological impression on humans that time is the rate at which our own organism runs. This last rate depends on the electron mass in the frame in which we are located. Consequently, we must get familiar with the phrase "difference of clock displays" (ACDframe) remembering that it corresponds to the "time interval" believed to be felt by an observer in that particular frame.
We have seen above that two clocks located in different gravitational potentials will not show the same difference of clock displays during the same real time interval. We will see now that quantum mechanics also predicts that clock rates are different when these clocks are carried in frames having different kinetic energy. We might assume that the relativistic correction could be made simply by taking into account the increase of electron mass due to the addition of kinetic energy, but this correction is too simple and incomplete (as we will see in sections 2.8 and 2.9) and disregards the need to consider the transfer of internal excitation energy between systems. In order to be able to calculate relative clock rates, we must first find the relationship between the excitation energy of atoms in frames having different velocities.
2.4 - Description of the Reference Meter.
The standard definition of length uses a unit called the "meter". In order to be coherent, we must define the meter in a way that can be reproduced in any frame. It is generally believed in physics that one can transfer, without any change of length, a standard meter from the rest frame to the moving frame. This is wrong because this is not compatible with the principle of mass-energy conservation and with quantum mechanics. When kinetic energy

CHAPTER TWO. TRANSFORMATION OF EXCITATION Energy BETWEEN FRAMES. 35

(or potential energy) is added to or removed from a rod, the electron mass and the Bohr radius change as required by the principle of mass-energy conservation. Consequently, the length of a rod will not be the same in frames having different velocities. The change of length of a standard rod which is one meter long in an initial frame can be calculated considering its kinetic and potential energies.
Even the most fundamental definition of the meter (which is 1/299 792 458 of the distance traveled by light in one second) suffers from the same error since it requires the use of the unit of time and since the "apparent second" in the moving frame (ACD(S)[mov]) is different from the "apparent second" in the rest frame (ACD(S)[rest]) due to the change of mass of the electrons in the atomic clock carried by the moving system. Consequently, to be able to compare lengths in different frames, we must complete the international definition of the reference meter and state its potential and kinetic energies.
We define here that the length of the reference meter corresponds to 1/299 792 458 of the distance traveled by light during one second on a clock located at rest in outer space, far away from the Sun.

2.5 - Definition of the Velocity of Light.
We want to point out that none of the above definitions depends on the experimental measurement of the velocity of light. The value of the parameter c is defined in equation 1.3 from the fundamental concept requiring an absolute constant K of proportionality between mass and energy:

E = Km.

2.3

However, it has been observed experimentally that the value of

K is equal to the square of what is interpreted to be the velocity of

light. Whatever c is, for practical reasons, we define it as:

c = VK.

2.4

Everywhere in this book, the meaning of c is fundamentally bound to equation 2.4. We believe that the fact that the velocity of light is equal to the square root of the constant K in the mass-

36 CHAPTER TWO. Transformation of Excitation Energy between Frames.
energy relationship is not just a coincidence and results from a fundamental mechanism. However, it is very likely that the best method of measuring the mass-energy constant K is through the measurement of c.
2.6 - Need of Parameters with a Double Index.
From the above description, we realize that the observer's frame is submitted to several particular conditions like its gravitational potential and kinetic energies. However, an observer moving with his clock cannot measure the change of clock rate because all phenomena in the moving frame, including the clock rate, change in the same proportion.
The same can be said of masses. When an observer and some masses move at an identical velocity, the values of the masses (as measured by the observer inside the moving system) are indistinguishable from the values obtained before the common change of velocity. After claiming that a mass increases with velocity with respect to an observer at rest, it would be incoherent to claim that the same mass does not increase when the observer moves with it.
In order to make a clear and coherent description, one must use a suitable notation which gives a complete description of the units used. To do this, two independent indexes are necessary. The first index indicates the units used for the measurement. For example, we can measure the length of an object either with respect to a reference meter at rest or with respect to a moving meter. It must be realized that the reference meter at rest is a unit that has a different length than the same reference meter in motion. It is almost like using inches instead of centimeters. When we measure a length / and a mass m using the units of length and mass issued from the system at rest, the length is represented by /[rest] and the mass is represented by m[rest]. When we measure lengths and masses using the units of the system in motion, we represent the length by /[mov] and the mass by m[mov]. The indexes [rest] and [mov] do not tell us whether the mass is moving or not. They only tell us what units are used.

CHAPTER TWO. TRANSFORMATION OF EXCITATION ENERGY BETWEEN FRAMES. 37

The second index indicates the state of motion of the system on which parameters (like length or mass) are measured. We describe the frame in which the particle is located using the subscript "v" when the particle is moving and the subscript "s" when the particle is stationary. For example, the mass of a stationary particle (using units of the rest frame) is represented by ms[rest] and the mass of a moving particle (using units of the rest frame), by mv[rest]. According to relativity, we must write:

mv[rest] = yms[rest].

2.5

Similarly, the mass of a moving particle measured using moving units is represented by mv[mov] and the mass of a stationary particle measured using moving units is represented by ms[mov]. Consequently, the number of kilograms in ms[rest] is identical to the number in mv[mov] because they are both measured using proper parameters. However, the mass merest] is different from mv[rest] as seen in equation 2.5.

The number "n" of meters of a rod does not change when the rod is moved to another frame as long as we measure proper values (number of proper meters). Then ns equals nv. However, the distance between the atoms changes. Since the interatomic distance a changes when a physical body is moved to another frame, the number of atoms Ns along a length of one meter[rest] in a stationary rod is different from the number of atoms Nv along the same length (one meter[rest]) when the rod is in motion at velocity v. Therefore when measuring the same absolute constant length in two frames we find:

2.6 meter[rest] meter[rest]'

Of course, the indexes [rest] and [mov] are irrelevant with the numbers ns, nv, Ns and Nv because they are pure numbers.

The fundamental importance of the necessity of using a double

index must not be underestimated because relativity cannot be

explained properly without it. This is a consequence of having

different units of mass and length in different frames. These

double indices are irrelevant in Newtonian mechanics.

In

principle, a third index could be added giving the information

38 CHAPTER TWO. TRANSFORMATION OF EXCITATION ENERGY BETWEEN FRAMES.
about the gravitational potential energy. This third parameter will be considered separately.
2.7 - Apparent Lack of Compatibility for Fast Moving Particles.
When a body is accelerated, its mass increases according to the relationship given by equation 2.5. Therefore fast moving atoms possess more massive electrons. Using the Bohr equation, let us calculate the consequences of a heavier electron in the case of the hydrogen atom.
When the electron mass is larger and no other parameter is taken into account, then according to the Bohr equation (equation 1.12), all the atomic energy levels should have more energy (equation 1.13). Consequently, since E = hv, the atoms formed with those heavier electrons should emit electromagnetic radiation at a higher frequency v. This means that an atomic clock located in the moving frame should run at a higher rate. However, we know from experiments that fast moving particles disintegrate at a slower rate and atoms emit a lower frequency. This has been clearly observed in the muon's and spectroscopic experiments. We conclude that the increase of electron mass that causes atoms to disintegrate at a higher rate in a gravitational potential does not appear to be compatible with the slower rate of disintegration of fast moving muons. This apparent contradiction is a very serious problem that requires a more careful study. Using the principle of mass-energy conservation, we will solve that problem by showing that one important parameter has been ignored.
In the next section, we will consider solely experiments in which the gravitational potential energy is always constant. This corresponds to the study of special relativity. Only the velocity (and therefore the kinetic energy) will change. The problem of combining gravitational potential energy with kinetic energy will be studied in chapters five and six.

CHAPTER TWO. TRANSFORMATION OF EXCITATION ENERGY BETWEEN FRAMES. 39

2.8 - Demonstration of the Energy Relationship between Systems.
Let us consider a stationary particle Mso where the index s stands for stationary and the index o means that the particle is in its ground state of internal excitation. That particle can be a single hydrogen atom. When accelerated to a velocity v, its mass becomes:

Mvo[rest] - yMS0[rest]

2.7

where the index v means that the particle has a velocity v.

Let us consider that an internal energy of excitation Exs[rest] is given to that particle before its acceleration. The index x refers to internal excitation energy. The total mass Msxt[rest] of the stationary excited atom is then:

E

Msxt [rest] = Mso [rest]+ —^ [rest]

2.8

where the index t refers to the total mass-energy which includes rest mass, internal and kinetic energies when relevant. From equation 2.8, we calculate that the internal excitation energy Exs[rest] alone has a mass-equivalent Mxs[rest] given by:

E

hv

M xs [rest] = —f- [rest] = -f- [rest]

2.9

c

c

where hvs[rest] is the energy Exs measured using the units of time and length of the rest frame. Equations 2.8 and 2.9 give:

Msxt = MS0[rest]+Mxs[rest],

2.10

The particle of mass Msxt can emit its energy of excitation according to equation 2.9. When that particle (Msxt) is accelerated to a velocity v, its mass becomes Mvxt which is y times its mass at rest as given by equation 2.5. This gives:

Mvxt[rest] =yMsxt[rest].

Putting 2.10 in 2.11 gives: Mvxt[rest] = yMS0[rest]+yMxs[rest].

2.12

40 CHAPTER TWO. Transformation of Excitation Energy between Frames.

If the particle does not possess any internal energy, then the second term of equation 2.12 vanishes and we get equation 2.7. Putting equation 2.7 in 2.12, we have:

Mvxt[rest] = Mvo [rest]+7MXS [rest].

2.13

Equations 2.13 and 2.9 give:

yhvQ

Merest] = Mvo [rest] +

[rest].

2.14

Equation 2.13 shows that the velocity of the excited particle leads to the mass component Mvo[rest], The second term yMxs[rest] gives the mass-energy equivalent of the excitation energy of the moving particle. This term is composed of the mass equivalent of the excitation energy of the particle (which is hvs/c [rest]) and of the energy required to accelerate it (given by y). From equations 2.13 and 2.14, we see that the principle of mass-energy conservation requires that the total energy of excitation combined with the energy necessary to accelerate that energy of excitation (or its mass equivalent) give:

En(Excit.-(-acceleration of excit.) = yMxsc"[rest] = yhvs[rest].2.15

Equation 2.15 gives the total energy [rest] that the excited moving atom must lose (by emission of a photon) to go to its ground state.

Elowever, when the observer moves with the excited atom and uses rest units, he will deduce from his measurements a frequency vv[rest] from which he will naturally decide that the energy of internal excitation is hvv[rest]. Therefore:

En[rest](emitted) = hvjrest],

2.16

The energy that was required to accelerate the mass-equivalent of that excitation energy may appear irrelevant to the moving observer. However, due to mass-energy conservation, that energy cannot disappear and be ignored. According to the principle of mass-energy conservation, since no other photon is emitted during the transition, the emitted photon must possess all the energy available which includes the energy of excitation plus the kinetic energy of the mass equivalent of that excitation energy.

CHAPTER TWO. Transformation of Excitation Energy between Frames. 41

Using the same units, it is clear that the total energy of equation 2.15 (excitation plus the energy required to accelerate the massequivalent of the energy of excitation) is equal to the energy of the photon received during the de-excitation by the observer at rest (equation 2.16). This gives:

yMxsc2[rest] = yhvs[rest] = hvv[rest].

2.17

In equation 2.17, we have the Planck parameter h that comes from the measurement of hvs in a stationary frame. We also have the Planck parameter h that comes from a measurement of hvv in the moving frame (always using the same common units [rest]). In order to be coherent and since the Planck parameter comes from measurements from different frames, we must individually label each Planck parameter. Equation 2.17 becomes:

yhsvs[rest] = hvvv[rest].

2.18

Equation 2.18 is an important relationship that must be applied when the energy of excitation is given a new velocity.

2.9 - Relative Frequencies between Systems.
In order to solve equation 2.18, we need to find a relationship between vs[rest] and vv[rest]. Let us consider an electromagnetic wave of frequency vv[rest] emitted by an atom having a constant velocity v. That electromagnetic wave is measured by an observer in the rest frame. When the measurement of the frequency is made, he must consider two different phenomena that might change the frequency due to the velocity of the emitting atom. The first one is the change of clock rate of the emitter and the second is the classical Doppler effect due to the radial velocity between the stationary source of radiation and the moving observer. Let us study those two effects separately starting with the classical Doppler effect.
Let us suppose that the source of radiation moving at a velocity v is emitting in a direction perpendicular to its velocity. The observer at rest receives the radiation at a frequency vs[rest]. This special direction allows us to take the classical Doppler effect into account very easily. We know that the Doppler correction is given by the relationship:

42 CHAPTER TWO. TRANSFORMATION OF EXCITATION ENERGY BETWEEN FRAMES.

2.19

where vr is the radial velocity between the moving frame and the frame at rest. In the case of a tangential velocity, when vr = 0, there is no Doppler correction. Consequently:

1- — V cJ

2.20

Equation 2.20 in 2.19 shows that light received by the observer at rest from a moving source emitting radiation at 90° with respect to his velocity produces a Doppler effect equal to zero. At a different angle, when the Doppler effect is different from zero, we will see (chapter eight) that the change of frequency is completely due to the recoil produced by the emitting photon on the emitting particle. In that case, the recoil of the emitting particle gives to that emitting particle the energy lost or gained by the photon in agreement with mass-energy conservation.

Let us consider now the mechanism which changes the clock rate when particles move from a rest frame to a moving frame. We expect the frequency of the clock to change when moved from a rest frame to a moving frame. However, that consideration is totally irrelevant. It is true that a clock will emit a different frequency when it is carried to a moving frame but this is not the problem considered here. The problem here refers to a comparison between a frequency that is measured to be vv[rest] in the moving frame using the rest frame units and a frequency measured by an observer at rest vs[rest] (using also rest frame units). Of course, since the units are the same and there is no Doppler correction we have:

vv[rest] = vs[rest].

2.21

Equation 2.21 shows that using the units of the rest frame, the moving and the stationary observers will observe the same absolute frequency when a wave is traveling between systems which have no radial relative velocity.

Combining equations 2.21 and 2.18 gives:

hv[rest] = yhs[rest].

2.22

CHAPTER TWO. TRANSFORMATION OF EXCITATION ENERGY BETWEEN FRAMES. 43

Equation 2.22 means that when we use the Planck parameter h to determine the energy in a moving system we must make a correction (y) because of the kinetic energy of the equivalent mass ot the excitation energy hvvv[rest]. Keeping h identical between frames would be the same thing as claiming that masses do not change when they are accelerated.

Equation 2.22 is the relationship we were looking for in section 2.1. It is, for energy, the relationship equivalent to the mass-energy conservation principle:

mv[rest] = yms[rest],

2.23

Equation 2.22 is a relationship previously ignored. However this equation, which is required by the principle of mass-energy conservation, is absolutely necessary when treating problems dealing with a change of velocity of internal energy. We will see in chapter three how equation 2.22 allows us to solve the apparent contradiction described in section 2.7.

2.10 - Cases of Relevance of the Relationship hv=yhs.
We must notice that equation 2.22 (hv[rest] = yhs[rest]) results from the fact that the internal excitation energy of particles (that has a mass equivalent) acquires a velocity v that produces an increase of mass-energy equivalent. However, in the case of a change of gravitational potential energy, as seen in chapter one, the mass-equivalent of the internal excitation energy has no kinetic energy since it has no velocity. Therefore in the case of potential energy, the relationships hv[rest]=yhs[rest] and mv[rest] = yms[rest] are irrelevant since y = 1 when v = 0. In the case of gravitational potential, the changes of energy and length are given by equation 1.22 in chapter one.
Let us finally note that the relationship hv[rest] = yhs[rest] is absolutely necessary to satisfy the principle of invariance of physical laws in any frame of reference as will be seen in the rest of this book.

44 CHAPTER TWO. Transformation of Excitation Energy between Frames.

2.11 - Symbols and Variables.

ACDframe

difference of clock displays on a clock located in a

frame

ACD(S)[mov] ACD for the apparent second in the moving frame

ACD(S)[rest] ACD corresponding to the apparent second in the

rest frame

Exs[rest]

energy of excitation given at rest in rest units

hs[rest]

Planck parameter on the rest frame in rest units

hv[rest]

Planck parameter on the frame in motion in rest

units

ms[rest]

mass of an object at rest in rest units

MJrest]

mass of a particle at rest in its ground state in rest

units

HJrest]

mass of the excitation energy of a particle at rest in

rest units

Msxt[rest]

total mass of a particle at rest in its excited state in

rest units

mv[rest]

mass of an object moving at velocity v in rest units

Mvo[rest]

mass of a particle in motion in its ground state in

rest units

Mvxt[rest]

total mass of a particle in motion in its excited state

[rest units]

vs[rest]

frequency of light measured by an observer at rest in

rest units

vv[rest]

frequency of light measured by a moving observer

in rest units

Chapter Three Demonstration of the Lorentz Equations without Einstein’s Relativity Principles.
3.1 - Fundamental Physical Principle.
In this chapter, we will show that the Lorentz equations can be demonstrated using the principle of mass-energy conservation and quantum mechanics. The equations obtained are mathematically identical to the usual Lorentz transformations. There is no need for Einstein's relativity principles or for the hypothesis of the constancy of the velocity of light. In fact, no new physical principle is required and the constancy of the velocity of light appears as a consequence to mass-energy conservation.
We have seen in chapter one that the principle of mass-energy conservation implies that the mass of a particle changes with the gravitational potential. In this chapter, we will consider particles with kinetic energy. We will take into account that masses increase with kinetic energy, using Einstein's relativistic relationship mv[rest] = yms[rest]. This relationship shows that a moving particle has a larger mass than the same particle at rest (using rest mass units). However, as expected, when observed within the moving frame (using proper values), the mass does not appear to change.
In order to demonstrate the Lorentz equations using physical considerations instead of a mathematical transformation of coordinates, we must define accurately the physical meaning of the quantities used. We have seen that Einstein considered that time is what clocks display. We know that clocks run more slowly when they are located in a gravitational potential. However, time does not flow more slowly because clocks run at a slower rate.
Consequently, even if the equations that we will find are mathematically the same as the Lorentz equations, because of Einstein's interpretation, the parameter representing the time t in the equation will actually be a clock display CD. Therefore due to Einstein's confusion between clock display and time, the units (second) characterizing time t in Lorentz's equations should not exist because t is actually a clock display (which is a pure number).

46 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...

When we compare Einstein's model of time dilation with the natural explanation in which the clock rate is simply slower, we are obliged to compare clock displays, which have no units, with real time, which needs to be expressed in seconds. In this chapter, since we wish to establish a comparison between Einstein's model and mass-energy conservation, it is impossible to avoid momentarily giving Einstein's units of time to quantities that represent only clock displays. Furthermore, we see that the relationship in which length / equals velocity times a time interval (/ = vAt), leads to an erroneous length because Einstein's definition of time is not time but a clock display. Therefore the length found is not a length but a pure number (of local meters). The length of a rod is a reality independent of the observer and does not depend on the rate at which a measuring clock is running. There is no change of length of a rod when the observer uses a clock running more slowly. Consequently, comparing our calculations with Einstein’s theory is very subtle because Einstein confused the slowing down of clocks with time dilation.

3.2 - Change of Energy and Bohr Radius Due to Kinetic Energy.
We have explained that the Bohr equation (equation 1.12) gives a relationship between the parameters that describe the rate at which an atomic clock runs. The energy levels in the Bohr atom for each of the n quantum levels are:

27i2k2e4

En,o[rest]

InI 2 nl02 mjrest]

3.1

where the subscript o means that the atom is at rest. When the hydrogen atom is given a velocity, the energy of each of the n levels changes as seen by an observer remaining at rest and using rest units.

We must notice that the frame in which the observer is actually located has no physical relevance. However, a description of the units (of mass, length and clock rate) used by the observer is necessary. Of course, one generally assumes that the observer uses the units that exist in his own frame. However, the description will be complete only when we specify the frame of origin of the units

CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY... 47

instead of assuming every time that the observer uses the units of his own frame.
The energy levels of the moving atom (using rest frame units) are given by putting equations 2.22 and 2.23 in equation 3.1. The Bohr equation becomes:

1 27i2k2e4

E [rest] =-2T2 m0[rest].

3.2

Y n h„

Furthermore, since the Bohr radius an of an atom at rest is:

a0[rest] =

[rest]

3.3

47t2m„e2k

using equations 2.22, 2.23 and 3.3, the Bohr radius of a moving atom will be:

av [rest] = y - - - [rest] = ya [rest].

3.4

47i m0e k

This means that the Bohr radius a0 increases linearly with y. This will be discussed in section 3.4. From equation 3.2, we see that the energy between atomic transitions of a moving atom (which determines the clock rate) decreases linearly as y increases (using the units of the rest frame). We conclude that according to quantum mechanics, the rate of a moving clock slows down when its velocity increases.

This is compatible with the slower clock rate of moving atoms as observed experimentally and interpreted erroneously as time dilation. The popular phrase "time dilation" should be interpreted as meaning that the rate of the moving clock has slowed down and not that time has dilated. Combining the Bohr equation (equation 3.2) with solely the mass relationship (equation 2.23) and neglecting equation 2.22 would lead to a rate increase of the moving clock. This is contrary to observations and to mass-energy conservation, as seen in chapter two. The correction due to massenergy must be applied to the Planck parameter h as given by equation 2.22. Consequently, the observed slowing down of the clock rate of moving clocks, which is implied by equation 3.2, is an

48 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...

experimental confirmation of equation 2.22. This also solves the apparent contradiction presented in section 2.7.

3.3 - The Lorentz Equation for Time.
From the relativistic Bohr equation presented above, let us calculate the energy of an atom located on a stationary frame. From equation 3.1 we see that the energy states of a stationary atom (using rest frame units) are:

2rr2k2e4

E0[rest] =-2T2— mo[restl = hovo[restl

3-5

n h0

where h0v0[rest] is the internal energy of excitation in the atom, using rest frame units. Due to its velocity, the atom located on the moving frame has a different internal energy. Equation 3.2 gives (using rest frame units):

Ev[rest] =

-1 ■ Y

271 2ki,2e4 n 2hu2

m0 [ rest]

=

h0vv[ rest]

3.6

where h0vv[rest] is the internal energy of excitation of the moving atom (using rest frame units) that can possibly be received on a frame at rest in order to be compatible with mass-energy conservation. Consequently, the radiation emitted from such an atom has a lower absolute energy and frequency. This can be seen from equations 3.5 and 3.6:

E0[rest]

Ev[rest]

3.7

y

From equation 3.7, we see that using rest units, there is less

internal energy Ev[rest] in the moving atom (due to equation 2.22)

than in the atom at rest (E0[rest]).

The middle term of equation 3.6 represents the internal excitation energy of the moving atom in rest units while the right hand side term represents the same internal energy available that can be received by an observer at rest (also in rest units). Since the energy states of the moving atom have less energy (always in rest units), the observer at rest will detect a lower frequency (as measured using rest frame units) if that energy is emitted. We must notice that in both cases (equations 3.5 and 3.6), the constant h

CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY... 49

refers to a measurement done in the stationary frame (meaning that the measurement is made from a frame having zero velocity and using rest units) so that the parameter h must have the subscript o.

One must notice a fundamental physical mechanism implied in the decrease of internal energy in the hydrogen atom as given in equation 3.7 (using rest units). The internal potential energy in a hydrogen atom is given by equation 1.12. When the hydrogen atom is moving, equation 1.12 shows that due to the increase of velocity, the electron mass me and therefore the energy En increases by a factor y. However, at the same time, the Planck parameter which is squared and located at the denominator also increases. The overall effect is that the internal energy En in the atom decreases when the velocity increases. One must then realize that when the velocity increases, the electron mass becomes larger but the decrease of the Planck parameter corresponds to a decrease of the force between the electron and the proton.

From equations 3.5, 3.6 and 3.7 we obtain that the ratio between the clock rates of the moving clock and the clock at rest is:

Ev[rest] _ h0vv[rest] _ 1 _ vv[restj

3g

E0[rest] h0v0[rest] y v0[rest]'

The last term vv[rest]/v0[rest] of equation 3.8 gives the ratio between the frequencies (in rest units) of oscillation of two independent clocks having different velocities according to the Bohr equation. This relationship has nothing to do with the relative values of the frequencies of an electromagnetic wave as given in equation 2.21. In equation 3.8, there are two different frequencies emitted by two different clocks observed in a single frame. However, in the case of equation 2.21, we have a single clock emitting a single frequency observed by two independent observers located in different frames.

Let us consider figure 3.1 on which a moving clock M travels in front of a station (at rest) from A to B. Let us measure the difference of clock displays ACD0 recorded on a clock located on the station at rest between the instants the moving clock M passes from A to B. We will also measure the difference of clock displays ACDV recorded on the moving clock while it passes from A to B. It

50 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...
is clear that the absolute time (as defined in section 2.3) is the same for M to pass from A to be B in both observations.

Station >'///////////7777? <--e-> Figure 3.1

However the two clocks will not display the same difference because they do not run at the same rate. The ratio between those two differences of clock displays ACD0 and ACDV is proportional to the ratio of the clock rates vjrest] and vv[rest]. Therefore:

ACDV vv[rest]

acd0

3-9

Combining equation 3.9 with equation 3.8 gives:

ACP0

ACDV

y

which is mathematically identical to:

3.10

ACDV =^yACD0.

3.11

From the usual definition of y, equation 2.2:

1

v2

1 y c — = - —

we find that, using equation 3.11:

ACDV = y V

.23
ACD0. c )

3.12 3.13

Einstein made the hypothesis that "time is what clocks are measuring". This means that the At in Einstein's relativity and in the Lorentz equations is only a difference of clock displays on a clock at rest to which the units of time were given:

ACD0 = At.

3.14

CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY... 51

In reality, since At is nothing more than a ACD, the units of ACD0 (which is a pure number) must be given to At. Let us give an example. It is believed that in Einstein's relativity and in the Lorentz equations, when an excited atomic state of a moving atom has not become de-excited after a classical time interval, it is because the time interval was shorter within the moving frame than in the rest frame. We have seen above that this explanation is incorrect and that the reason is that the principle of mass-energy conservation requires a change in the atom parameters and consequently, a slower internal motion inside atoms. This slower internal motion makes moving clocks function more slowly. Therefore, the At measured by Einstein's and Lorentz's clocks is not a time interval at all, but a difference of clock displays (ACD) of a clock running more slowly. The correct explanation is that when, in the Lorentz equation, we find that the At' is different from At during the same time interval, we are fooled by clocks running at different rates in different frames. It is an error of interpretation to give time units to At and At' in the Lorentz equations while they are no more than differences of clock displays as admitted by Einstein. Since the ACD is a pure number, the At in equation 3.14 is also a pure number. Similarly, the difference of clock displays ACDV is called At' in the Lorentz equations:

ACDV = At'.

3.15

A comparison with the Lorentz equations, as given with equations 3.14 and 3.15, is useful to examine some mathematical properties common to both interpretations. Equations 3.14 and 3.15 in equation 3.13 give:

3.16

By definition, the number of units x representing the distance

traveled during At (for Einstein corresponding to the time while a

clock shows ACD0) is:

x = vAt or x = vACD0.

3.17

52 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...

Of course, x is not a real distance, as explained in section 3.1. Let us substitute At from equation 3.17 to the second term At of equation 3.16. We get:

f vx

(

vx^

At' = y At - —5- or ACDV = y ACD - -y

k c)

k

c J

Equation 3.18 gives the relationship between At' (which is a difference of clock displays) displayed by a clock located at a distance x from the origin and moving at a velocity v and At displayed by a stationary clock. We observe that equation 3.18 is exactly the Lorentz equation for time and that it is compatible with Einstein's hypothesis that time is what clocks display. This equation is simply an exact mathematical description of massenergy conservation in agreement with equations 2.22 and 2.23 and with the physical mechanism implied by equation 3.2 We notice finally that the Lorentz transformation for time has been demonstrated here without using the hypothesis of the constancy of the velocity of light nor any new hypothesis. We have used only the mass-energy relationship E — Km from equation 2.3. In fact, we have obtained the Lorentz equation for time without the use of any of Einstein's relativity principles.

One must conclude that the Lorentz transformation derived above is in reality a transformation of relative clock displays between frames. Then At and At' (when related to this Lorentz equation) represent differences of clock displays ACD.

3.4 - Length Dilation Due to Kinetic Energy.
Length dilation and contraction have been demonstrated in chapter one for matter placed in a gravitational potential. Using equation 3.4, we will now show that the Bohr equation also gives a change of length when matter acquires a velocity v. This will be done without involving the constancy of the velocity of light. According to equation 3.4, we have:

ho

av

[rest]

=

y

- — —~2

5

[rest]

=

y«0

[rest].

4tc m0e k

3.19

Therefore, the relative size of the Bohr radius as a function of velocity is:

CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY... 53

tfjrest]
-= y
a0[rest]

3.20

Let us consider a reference meter made of ordinary classical atoms. We see from equation 3.20 that the size of atoms, which is proportional to the Bohr radius or to the interatomic distance (see Appendix I), increases as a function of velocity. This means that the size of all material matter increases with velocity.

We know that the number of atoms Na making up the length of a rod does not change with velocity. Furthermore, it is well established in modem physics that the interatomic distance (p0 is proportional to the Bohr radius a0 so that (pv[rest] = y(p0[rest], The length l0 of a rod is:

/0[rest] = (Na-1 )cp0[rest].

3.21

At velocity v, the length /v is:

/v[rest] = (Na-l)(pv[rest] = (Na-l)y(p0[rest],

3.22

We note that the number of atoms Na is much larger than unity. Therefore, using equations 3.21 and 3.22 we have:

/0[rest] /v [rest] = y/0 [rest] =

3.23

Equation 3.23 shows that there is length dilation of matter when its velocity increases (in a constant gravitational potential). Length dilation is a real physical phenomenon involving no stress nor any pressure, similar to length dilation and length contraction in a gravitational field, as shown in chapter one. It is just the natural equilibrium of matter given by quantum mechanics that makes it dilate at relativistic velocities. Space dilation or space contraction is meaningless.
The fact that we are led from our reasoning to length dilation instead of length contraction does not represent a problem since the assumed phenomenon of length contraction has never been observed experimentally in special relativity. On the contrary, we need length dilation to be compatible with the slowing down of clocks, which is also required by quantum mechanics and has been

54 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...

observed experimentally. In order to be coherent with quantum mechanics and mass-energy conservation, one must understand that there exists no length (nor space) contraction in special relativity because y is always equal to or larger than one (equation 3.23). Only length dilation can be produced when there is an increase of velocity.

3.5 - The Lorentz Transformation for Lengths.
Let us consider two identical frames O-X at rest. The axis of those frames are constructed with many rods in series each having a length exactly equal to one reference meter (defined in section 2.4). A mass M is located at a distance x[rest] from the origin 0[rest]. For a stationary observer using the reference meters located on the frame at rest, the coordinate of the mass M is:

x[rest] = n0meter[rest]

3.24

where n0 is the number of times the meter rod, when defined at rest (meter[rest]) must be used to form the length x[rest]. The symbol n0 is a pure number measured in the stationary (subscript o) frame. We must recall that contrary to Newtonian physics, the simple use of the number n0 is not sufficient to represent a length. A length must necessarily be represented by a pure number multiplied by the length of the reference meter.

Let us give the velocity V to one of the frames that we now call O'-X'. At time t = 0, the origin O' of the moving frame coincides with the origin O of the rest frame. The axis O'-X' is arbitrarily displaced on figure 3.2 in order to avoid confusion. Before the frame O'-X' acquired its velocity, the distance between the origin O and the mass M was identical in both systems. After the frame O'X' has reached velocity V, we have seen that the Bohr radius and all physical material on the moving frame are dilated as given by equation 3.23. Therefore the reference meters used to form the axis are longer. The mass M' on the moving frame is fixed with respect to that frame and does not move with respect to the particular segment of meter where it is fixed. Therefore the number nv of those standard moving rods between M' and the origin O' is necessarily the same and n0 = nv.

CHAPTER THREE, the Lorentz Equations without Einstein’s Relativity... 55

O'fmov]
Ofrest] _ <-x[rest]

-x'[mov] ■

M* -> X-

> X

Figure 3.2

However, the absolute distance x'[mov] between M' and O' will increase because the length of the standard meter has increased due to the increase of the Bohr radius. The distance x'[mov] between M'[mov] and the origin O' is given by:

x'[mov] - nvmeter[mov] = n0meter[mov]

3.25

with

nv = n0.

3.26

Using the notation x[rest] = /0[rest] and x'[rest] = /v[rest] equation 3.23 gives:

x'[rest] = yxfrest] or Ax'[rest] = yAx[rest].

3.27

Equation 3.27 means that using rest frame units, the distance x' (which is O'-M') is y times longer than the distance x (which is OM) also using rest frame units even if the numbers of local meters n0 and nv are the same.

3.5.1 - Apparent and Absolute Time.
In order to predict the consequences of the change of "clock" rate between systems, we must be able to compare predictions between different frames. Let us examine the relationship between the "apparent time" in different frames. In Einstein's relativity, the "time" is defined as what is perceived by each observer. It is equal to what a clock measures in its own frame. It is called t in the rest frame and t' in the moving frame.

56 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...

Consequently, each frame has its own "time" but we know that it is only apparent. Real physical time does not flow faster because the local clock runs faster. For an observer at rest, Einstein's interpretation assumes that his "time" t is the one shown by his clock at rest. Similarly, the "time" f is the apparent time in the moving frame. Since the moving clock runs at a different rate than the clock at rest (see equation 3.8), the time on the moving frame "appears" (as seen by an observer at rest) to elapse at a different rate giving:

t*f.

3.28

We define the "absolute second" S0[rest] as the time interval t taken by the atomic clock at rest (located away from any gravitational potential) to record a constant number Ns of oscillations. Since that clock at rest runs at a frequency v0[rest], the apparent rest second (called absolute second) will be elapsed when S0 equals unity. This gives:

Ns S0[rest]
vjrest]'

3.29

On a moving frame, the "apparent second" Sv[mov] is equal to the time taken by the local clock moving at velocity V to record the same number of oscillations Ns. Therefore during one "apparent second" (Sv) on the moving frame (at velocity V), by definition, the clock must record the same number of oscillations as the clock on the rest frame does during one "absolute second" (S0). This means that during one "apparent second" inside any frame, the local ACD is always the same number. Then, since clocks have different rates, in different frames, the "absolute duration" of the "apparent second" varies with the velocity of the frame carrying the clock.

It is arbitrarily decided that the rest second (in zero gravitational potential) is called the "absolute second of reference". Since the number of oscillations is the same for any local second, we have, for the case of apparent second Sv in a frame moving at velocity:

ACD(S0)[rest] = ACD(Sv)[mov],

3.30

CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY... 57

From the definition of apparent seconds in a frame moving at velocity V, with equations 3.29 and 3.30, we find that the duration of one moving second is:

Ns S v [rest]
vv[rest]

3.31

In order to be able to compare "apparent seconds" generated in different frames, we must be able to express the "apparent time" duration using common units. We have from equation 3.8:

v0[rest] = yvv[rest],

3.32

Equation 3.32 in equations 3.31 and 3.29 gives:

S v [rest] = y

= yS0 [rest].

v0[rest]

3.33

Equation 3.33 shows that the unit of time Sv in the moving frame is y times longer than the unit of time S0 in the rest frame.

Let us consider the "real time intervals" corresponding to the same numerical value of local apparent "x" seconds elapsed in both the rest frame and the moving frame. The ACD shown by either clock is the same in both frames. In Einstein's relativity, this was erroneously interpreted as the same time interval in both frames. In the rest frame, the real time t[rest] is equal to the number of seconds "x" times the duration of the apparent second S0 at rest. This gives:

t[rest] = xS0[rest],

3.34

In the moving frame, the real time (in rest units) is called t'[rest]. It is equal to the number "x" of seconds times the duration of the apparent moving second Sv:

t'[rest] = xSv[rest],

3.35

Combining equations 3.33, 3.34 and 3.35 gives:

f [rest] = yt[rest] or At'[rest] = yAt [rest],

3.36

Equation 3.36 shows that when we consider the same number of local "apparent seconds" (i.e. the same difference of clock displays) in two different frames, the real absolute time spent on the moving

58 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...

frame is y times longer that the absolute time spent on the rest frame.
Equation 3.36 is equivalent to equation 3.18 when time is measured at the same location (x = 0). However, one must understand that the change of time between systems suggested by Einstein is only apparent because clocks in different frames run at different rates. This has erroneously been interpreted as time dilation in the past, but we see now that it is nothing else than clocks running at different rates in different frames.

3.5.2 - Relationship between Velocities V and V'.
On figure 3.2, the right hand side direction of the axes O-X and O'-X' is positive in both frames. When the moving frame O'-X' has a velocity toward the right hand side, the coordinate of the location M' increases (in time) with respect to the rest frame O-X. Therefore location M' has a positive velocity with respect to the rest frame O-X. However, figure 3.2 shows that when the moving frame (with origin O') travels to the right hand side, location M moves to the left hand side with respect to the frame O'-X'. The coordinate of location M is getting more and more negative (in time) with respect to the frame O'-X', while the coordinate of location M' is getting more positive in time with respect to the frame O-X. This means that the velocity V' of point M' (with respect to O-X) has the opposite sign of the velocity V of point M with respect to O'-X'. This result comes out of pure geometrical considerations illustrated on figure 3.2. Therefore:

V

V'

— i vi = -— iv-r

3 37

Equation 3.37 signifies that the velocities have opposite directions. We will show now that the velocities V and V' have the same magnitude.

3.5.3 - Relative Velocities within Systems.
Let us consider a rest frame and a moving frame. Both frames were identical before the moving frame started to move at velocity V[rest], Inside both frames, we consider rods that had the same length when they were initially in the same frame at rest. This can

CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY... 59

be verified later if we count the same number of atoms in both frames for the length of either rods. The rod at rest extends from O to M and the moving rod extends from O' to M'.

There are at least two different ways to compare velocities between frames. One way consists of measuring directly the velocity in each frame using proper values and comparing numbers. Another way, the one we will use here, is to use a definition of velocity in each frame and to compare the corresponding elements of the definitions. The velocity u of a moving object across O-M with respect to the rest frame is defined as:

Ax[rest] u[rest] -
Atfrest]

3.38

With equation 3.38, we start dealing with a series of equations related to velocities. These velocities can have any direction in space and might be described by vectors. However, such a description would lead to a very heavy notation that could be confusing and would require useless efforts. This is avoided by defining that in every equation between 3.38 and 3.46, we consider that u and u' represent the magnitudes |u| and |uj of these parameters. The appropriate mathematical sign of the velocities will be considered starting with equation 3.46.

Inside the moving frame, a similar slowly moving object moves from O' to M' (distance Ax'). During the time At' the slow moving object crosses the distance Ax' from O' to M'. The velocity of the slow moving object with respect to the moving frame is defined as:

Ax'[mov] u'[mov] -
At'[mov]

3.39

We have seen that, before the moving rod (O'-M1) started to move, it was similar to the rod in the rest frame (O-M) and that both clock rates were similar. Consequently, we can use equations 3.27 and 3.36. Let us put the transformation of coordinates given by equations 3.27 and 3.36 into the equation 3.38. We get:

Ax[rest] yAx'[rest] _ Ax([rest] unrest] - ^tfrest] - yAt'[rest] ~ At'[rest]‘

3.40

60 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...

Let us use equation 3.23 to calculate the ratio between the units of length. If the length l0 is a unit of length equal to one meter using rest units, we see that this unit of length becomes yl0 on the moving frame. Therefore the relationship between the units of length is:

/0[mov] = y/0[rest] or meter[mov] = ymeterjrest], 3.41

This means that when we move from the rest frame to the moving frame, the unit of length becomes y times longer. Therefore, in order to represent the same physical length using longer units of length, the number of units Ax'[mov] must be smaller. This gives:

Ax' [rest]

Ax' [mov] =

y

3.42

In the case of time, a corresponding phenomenon takes place.

Let us consider equation 3.36. We see that a time interval At0 equal

to one unit of time in the rest frame becomes y times larger in the

moving frame because it takes more time for the slower clock to

show the same ACD. In that case, we see from equation 3.36 that

the change of local units of time At0 between frames gives:

At0[mov] = yAt0[rest] or sec[mov] = ysec[rest], 3.43

This means that when we move from the rest frame to the moving frame, the local unit of time becomes y times larger. Therefore in order to represent the same absolute time interval using longer units of time, the number of units At'[mov] must be smaller. This gives:

At'[rest] At'[mov]
y
Equations 3.39, 3.40, 3.42 and 3.44 give:

3.44

Axfrest] Ax'[rest] Ax'[mov] u[r6St] = At[rest] = At'[rest] = At'[mov] u'[mov], 3.45

Equation 3.45 shows that the velocity u measured using the rest frame units is the same as the velocity u' using the moving frame units.

CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY... 61

Among the values of velocities which can be given to u, we can choose the velocity V which is the velocity of the moving frame with respect to the rest frame (rest frame units). Symmetrically, let us call V' the velocity u' of the rest frame with respect to the moving frame (using moving frame units). Using equations 3.37 and 3.45 gives:

v = -v

3.46

V0[rest] = -Vv[mov],

3.47

Equation 3.46 shows that the proper value of the velocity of the moving frame with respect to the rest frame is the same (negative) as the proper value of the velocity of the rest frame with respect to the moving frame.

Let us add that a velocity appears as a physical concept for a physicist. However, we have seen above that a comparison of velocities in two different frames having a relative velocity leads to the same numbers. We have seen that when we are in a moving frame, the ratio between the distance traveled and the time taken to travel it changes with respect to the rest frame. Both the numerator (the distance) and the denominator (time interval) change by the same ratio. Consequently, a constant velocity is nothing more that a constant ratio between two fundamental physical quantities. One can say that the constant velocity in different frames means the same thing as three oranges out of six is the same thing as four apples out of eight. Velocities are just ratios of physical quantities.

3.5.4 - Lorentz’s Second Relationship.
In order to find the dynamical relationship between the coordinates x' and x, let us now combine the quantities x, V and t calculated above. In classical mechanics inside the moving frame we have:

x' = x0'+V'f

3.48

where x0' is the coordinate x at t = 0 and V' is the velocity between frames. In order to be more specific, in complete notation, equation 3.48 should be:

62 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...

xv[mov] = xov[mov]+Vv[mov]tv[mov].

3.49

Let us consider first in equation 3.49 the expression tjmov]. The term tv represents the number of units that is multiplied by the length of the unit [mov]. Let us calculate what would be the quantity tv[mov] using the [rest] units of length instead of the [mov] units of length.

From equation 3.44, we have: tv[rest] = ytv[mov],

3.50

In the case of the units of distance (xv or xov) we use again the same method. With the help of equation 3.42 we find:

xv[rest] = yxv[mov]

3.51

and Xovtrest] =yxov[mov].

3.52

From equation 3.49, transforming xv[mov] with 3.51, xOY[mov] with 3.52, and tjmov] with 3.50, we get after multiplying both sides by y:

xv[rest] = xov[rest]+Vv[mov]tv[rest],

3.53

From equation 3.53, transforming xov[rest] with 3.27, Vv[mov] with 3.47, and tv[rest] with 3.36, we get:

xv[rest] — y(x00[rest]-V0[rest]t0[rest]).

3.54

Using a more conventional notation this is:

x' = y(x-Vt).

3.55

Equation 3.55 gives the relationship between the coordinate x' on the moving frame and the coordinate x, the velocity V and the time t on the rest frame. This relationship results solely from massenergy conservation and quantum mechanics without using any of Einstein's relativity principles. However, equation 3.55 is exactly identical to the Lorentz equation related to lengths. The demonstrations leading to equations 3.18 and 3 55 show the uselessness of Einstein's special relativity principles. Most importantly, this demonstration provides a way to give a logical interpretation to experiments without space or time contraction or dilation.

CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY... 63

3.6 - Constant Velocity of Light within Any Frame of Reference.
We must notice that c is also a velocity obtained from the quotient of a distance by time within any frame. Let us consider that the internal velocity u is the velocity of light c. In the moving frame, the velocity u' equals c'. Therefore when the velocities u and u' considered are applied to light, equation 3.45 gives:

c = c\

3.56

When we use the complete notation, we get:

cv [mov] = cjrest].

3.57

This means that following equations 3.45 and 3.56, one must conclude that the physical mechanism resulting from mass-energy conservation and quantum mechanics leads to the conclusion (not the hypothesis) that any velocity, including the velocity of light, is constant as measured within any frame (using proper values). Contrary to Einstein and Lorentz, we do not have to make the arbitrary hypothesis that the velocity of light is constant inside all frames. We have found that the constancy of the velocity of light is a necessary conclusion to mass-energy conservation and the quantum mechanical equations.

From another point of view, the value of c, called the velocity of light, has been defined in section 2.4 as the square root of K (the quotient between energy and mass) which is the fundamental basis of mass-energy equivalence. Any theory or experiment not compatible with the constancy of the velocity of light (using proper values) is therefore necessarily not compatible with quantum mechanics and mass-energy conservation. However, since the velocity of light is given as the quotient of two quantities (length and ACD) that are different in different frames, the physical meaning of that constant ratio is subtle.

3.7 - Non-Reality of Space Dilation, Contraction or Distortion.
The distance Ax traveled in a time interval At is defined as:

64 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...

Ax = vAt.

3.58

Let us assume an observer traveling between the ends of a long stationary rod having a length Ax. That length Ax is calculated from the velocity v times the time interval At necessary to travel between the ends of the rod. We know that the velocity v is the same on any frame. However, the difference of clock displays ACD0 (which is interpreted as time At by Einstein) on the rest frame is different from ACDV (interpreted by Einstein as time interval At') on the moving frame. Consequently, according to Einstein's interpretation, the length Ax' measured by the moving observer is different from the length Ax of the same rod measured by the observer at rest. At the velocity of light, the ACDc decreases to zero so that the (apparent Einstein's) length Ax' becomes zero for the moving observer because his moving clock has stopped running.

It is irrational to claim that the length of the stationary rod changes and even becomes zero just because the observer changes his velocity. How can the length of a rod logically change because a non interacting observer looks at it? The rod would become longer or shorter depending on the observer's own velocity. The length (and other properties) of the rod would not be a property that would belong to matter. It is the observer that would set the length of the rod and different observers would simultaneously find different lengths for the same rod depending on their observing conditions. Then, what would be the length of the rod if there were no observer? It is just like the statement that the moon is not there when nobody is looking at it. We believe that this is nonsense and that the length of matter is independent of the observer. This is the same irrationality that appears in quantum mechanics and which has already been discussed [1],

We have not yet defined how to measure space. This is because space is not measurable unless we fill it up at least partially with matter. Then, it is that matter that we measure, not space. Whether space is empty or full of matter, we generally refer to it as "space". We know several methods of measuring lengths of objects but there does not exist any method of measuring space without using matter as a reference. In relativity, space is often referred to as

CHAPTER THREE, THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY... 65
being contracted or dilated. How can it be contracted or dilated when there is no method of measuring it without assuming some matter in it? The properties of matter are then inadvertently attributed to or confused with space. The same comment applies to the belief of space distortion. How can there be space distortion when we cannot measure space directly in the absence of matter? The interpretation of space distortion is nothing more than a change of the Bohr radius in the measuring instrument or in the matter filling the space.
This problem is easily solved logically when we consider that the internal atomic mechanism of the observer runs at a different rate since electrons in motion have a different mass. This has nothing to do with the illusion of space dilation or distortion. One must conclude that the expressions "space contraction" and "space distortion" are irrational. They bring confusion and must be eliminated.
3.8 - Transformation of Units in Different Frames.
There are many other consequences to the relativistic changes of lengths and masses. For example, in chapter one we have seen that the mass of particles decreases when located at rest in a lower gravitational potential. In chapter three we have seen that masses increase with velocity due to the absorption of kinetic energy. This means that if we take an object of one kilogram on Earth and move it to a location at rest on the solar surface, about one millionth of its mass will disappear and be carried away by the energy generated during the slowing down of the object falling into the Sun. Even if there is exactly the same number of atoms in one Earth kilogram after it is carried on the Sun's surface, we see that the solar kilogram has less mass than the Earth kilogram using any common frame of comparison of mass units. Consequently, there is more energy (in Earth joules) in one Earth kilogram than in one solar kilogram. This is required by the principle of mass-energy conservation.
Similar considerations must be applied to most physical constants. Because of the principle of mass-energy conservation, the units must always be specified (kg[Earth], meter[Earth],

66 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...

joule[Earth], second[Earth]). However, the electric charge appears to be constant in any frame. This means that the ratio of the electron charge divided by the electron mass (e/m) is different in different frames. For example, e/m is smaller on Earth (when using Earth units) than on the solar surface (using Earth units). In order to be able to compare those quantities with the ones calculated in different frames, we must take into account the difference of gravitational potential or the difference of kinetic energy. To define accurately the reference kilogram, the reference meter, etc., we must know the exact altitude on Earth at which these units have been defined.

3.9 - Failure of the Reciprocity Principle.
We have studied above some of the differences existing between a frame in motion and a frame at rest. In a moving frame, clocks run at a slower rate, the Bohr radius is larger and so are masses because of their kinetic energy. Let us consider a body on the rest frame having a mass m0[rest]. Its total energy is:

E0[rest] = m0[rest]c2.

3.59

When m0[rest] is accelerated to velocity v0[rest] with respect to the rest frame, its mass becomes mv[rest]. We get:

v2 mv [rest] = ym0 [rest] = m0[rest]+ m0 —^-[rest]+...
2c

3.60

Equation 3.60 shows that the moving mass mv[rest] is larger than the rest mass m0[rest]:

mv[rest] > m0[rest],

3.61

Let us consider now a train moving at velocity v0[rest] carrying an observer and the mass mentioned above. The mass of the train, of the observer and of the body described above becomes y times larger than when at rest. However, since the units in the moving train have been modified by the same ratio y, the changes of mass, clock rate and length are undetectable to the moving observer, even if they are real. Inside the moving train, an observer using Einstein's reciprocity principle will claim that the object of mass mv[rest] is at rest with respect to him. He will thus call it M0[rest], Therefore:

CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY... 67

Mjrest] = mv[rest] = ym0[rest].

3.62

It is because we use Einstein's hypothesis of reciprocity that we write [rest] after M0 in equation 3.62, since Einstein's hypothesis assumes that the mass that has been transferred to the train is now at rest for the observer moving with the train. Furthermore, the symbol = used in equation 3.62 does not mean that we are defining a new quantity. The symbol = means that M0 is the same object in the same physical condition as mv[rest].

Now, the moving observer takes the object of mass M0[rest] (that is stationary with respect to him) and throws it at velocity vjrest] with respect to his moving train (considered at rest in his frame) in the direction opposite to the direction of motion of the train. According to Einstein's principle of reciprocity, the mass projected at velocity v0[rest] with respect to the moving frame acquires velocity and energy with respect to the moving frame (now considered at rest). Einstein's principle of reciprocity says that all frames are identical which means that mass M0[rest] increases when accelerated with respect to the train to become Mv[rest], In fact, the reciprocity principle implies that the passage of the object of mass M0[rest] from zero velocityJrest] to vjrest] (with respect to the train) increases its mass by y times, independently of the direction of the velocity of the mass with respect to the train. This gives:

MJrest] = yMJrest],

3.63

As expected from the relativity principle, equation 3.63 shows that mass MJrest] is larger than MJrest] giving:

MJrest] > MJrest].

3.64

A physical representation of these changes of velocity shows that the mass MJrest] now has zero velocity with respect to the rest frame. It is back at rest on the rest frame. Mass MJrest] is then physically undistinguishable from mass mjrest] since it is the very same object having the same zero velocity with respect to the same rest frame. Therefore physically, we must have:

Mjrest] = mjrest].

3.65

Combining equations 3.62, 3.63 and 3.65 gives:

68 CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY...

2
mjrest] = Mv[rest] = y mjrest],

3.66

Obviously, equation 3.66 is correct only if y equals unity so that the velocity must always be zero. This shows that the principle of reciprocity cannot be valid when we apply the principle of massenergy conservation. We must conclude that Einstein's reciprocity principle is not coherent.

Contrary to Einstein's claim, the energy given to a mass accelerated with respect to the train must depend on the direction of its velocity with respect to the direction of the velocity of the train. When the directions are opposite, the two velocities (whose magnitudes are equal) cancel out and the mass of the body must come back to its original value in the rest frame. Otherwise we would discover that atoms of matter having traveled to another frame would have a different mass after their return to the initial frame. We must conclude that two frames cannot be equivalent when there exists a relative motion between them.

3.10 - References.
[1] P. Marmet, Absurdities in Modem Physics: A Solution. ISBN 0-921272-15-4, Les Editions du Nordir, c/o R. Yergeau, 165 Waller, Ottawa, Ontario KIN 6N5, 144p. 1993.

3.11 - Symbols and Variables.

ajrest]

Bohr radius at rest in rest units

ajrest]

Bohr radius in motion in rest units

ACD0

difference of clock displays on a clock at rest

ACD(S0)[frame] ACD corresponding to an apparent second in

any frame

ACDV

difference of clock displays on a clock in

motion

En Jrest] = EJrest] energy of the Bohr atom at rest in state n in rest

units

En Jrest] = EJrest] energy of the Bohr atom in motion in state n in

rest units

hjrest]

Planck parameter on the rest frame in rest units

hjrest]

Planck parameter on the frame in motion in rest

units

CHAPTER THREE. THE LORENTZ EQUATIONS WITHOUT EINSTEIN’S RELATIVITY... 69

/0[rest] /v[rest] v0[rest] Ns
vv[rest] (S0)[rest] (Sv)[rest] u[rest]
u'[rest]
V = V0[rest]
V— Vv[mov]
x[rest] x'[mov]

length of a rod at rest in rest units length of a rod in motion in rest units clock rate of a clock at rest in rest units number of clock oscillations in an apparent second clock rate of a clock in motion in rest units definition of the absolute second in rest units duration of one moving second in rest units definition of the velocity in the rest frame in rest units definition of the velocity in the moving frame in rest units velocity of M with respect to the moving frame in rest units velocity of M' with respect to the rest frame in motion units distance between O and M in rest units distance between O' and M' in motion units

Chapter Four Fundamental Nature of the Mechanism Responsible for the Advance of the Perihelion of
Mercury.
4.1 Definition of the Absolute Standard Units (o.s.J.
In order to understand the mechanism responsible for the advance of the perihelion of Mercury, we need to explain the meaning of quantities such as an absolute standard of mass, time or length. The meaning of absolute standards is such that each of them must always represent the same and unique physical quantity in any frame. This condition is necessary since the absolute length of a rod does not change because it is measured from a different frame. This also applies to an absolute time interval and an absolute mass: they do not change when measured in different frames. However, an absolute length, time interval or mass can be described using different parameters (e.g. different units). One must conclude that lengths, time intervals and masses are absolute and exist independently of the observer. They never change as long as they remain within one constant frame. However, they appear to change with respect to an observer who moves to a different frame because they are then compared with new units located in a different frame.
In relativity, we always read an expression with respect to a frame "of reference". The phrase "of reference" gives the illusion that masses, lengths and clock rates really change as a function of the "reference" used to measure them. That there could be a real physical change of mass, length and clock rate because the observer uses a different "reference" does not make sense. This apparent change of length, clock rate or mass is simply due to the observer using different units of comparison. In this book, we avoid the words "of reference" because they are clearly misleading.
We have seen that when a rod changes frames, its absolute length changes. However, when an observer carrying his reference meter changes frames, the length of the rod that remains at rest corresponds to a different number of the observer's new reference

CHAPTER FOUR. Fundamental Mechanism Responsible for the Advance... 71
meter. When a rod changes frames, the change of its length is real as seen in chapters one and three. However, when the observer changes frames (with his reference meter) and the rod does not, there is only a change in the number of measured meters; the rod does not change. Consequently, the change of frame of the rod and the change of frame of the observer (carrying his reference meter) are not symmetrical.
4.2 - The Absolute Reference Meter.
The usual definition of the meter is 1/299 792 458 of the distance traveled by light during one second. The local clock is used to determine the second. We recall from section 2.4 that this definition is not absolute because it depends on the definition of the second which is a function of the local clock rate which changes from frame to frame.
Unfortunately, there is no direct way to reproduce an absolute meter within a randomly chosen frame. We have seen that carrying a piece of solid matter from one frame to another one (in which the potential or kinetic energy is different) leads to a change of the Bohr radius and consequently to a change in the dimensions of the piece of matter. However, a local meter can apparently be reproduced in any other frame using a solid meter previously calibrated in outer space and brought to the local frame. Of course, the absolute length of that local meter in the new frame will not be equal to its absolute length when it was in outer space because the potential and kinetic energies may change from frame to frame.
One can also reproduce a local meter in any frame by calculating 1/299 792 458 of the distance traveled by light in one local second. However, the duration of the local second must be corrected with respect to the reference clock-rate existing in outer space (with v = 0). It is illusionary to believe that absolute time and absolute length can be obtained in any frame just by carrying a reference atomic clock and a reference meter to the new frame.
We define the absolute reference meter (meter0 s) as the distance traveled by light during 1/299 792 458 of a second given by a clock located at rest in outer space away from any mass. The subscript o.s. defines where the meter is located. This unit of length is equal

72 CHAPTER FOUR. FUNDAMENTAL MECHANISM RESPONSIBLE FOR THE ADVANCE...

to a number B0 s times the length of the Bohr radius ao s in outer space. An absolute reference meter must have the same absolute physical length, independently of the frame where it is located (and of the frame where the observer is located). Consequently, an observer must make relevant corrections to his local meter to reproduce the absolute reference meter. The definition of the absolute reference meter is then:

metero.s. = B0,a0„

4-1

The absolute meter can be reproduced in any frame but it is defined with respect to a length in outer space. The constant Bo s (the inverse of the Bohr radius) is about 1.8897263><1010. Since the Bohr radius a varies with the electron mass (which changes with potential and kinetic energies), the constant number B0 s times the outer space Bohr radius a is not an absolute standard when the meter is not located in outer space. The Earth meter (meterE) is different from the absolute reference meter (meteros) because the Bohr radius is longer on Earth. The length of the Earth meter is:

meterE = B0 s aE.

4.2

We see that the length of a meter at a Mercury distance from the Sun is also different from the length of a meter in outer space or on Earth. Let us study the example of Mercury since we wish to predict a phenomenon taking place at the distance from the Sun where Mercury is orbiting. The length of the Mercury meter (meterM) is:

meterM = B0 s aM.

4.3

In order to avoid useless lengthy repetitions, we will shorten some of the descriptions. Instead of repeating that we refer to a location at the Mercury distance from the Sun which has zero orbital velocity, we will simply say "Mercury location" and the context will provide the supplementary information. The velocity component of Mercury will be considered separately later. All other parameters will be taken into account only later because they are not relevant in this chapter and would bring confusion. An absolute standard of reference will sometimes be called in short "absolute meter", "absolute time rate" or "absolute mass" when it corresponds to the standard established in outer space.

CHAPTER FOUR. Fundamental Mechanism Responsible for the Advance... 73

In the problems considered in these first chapters, the relative changes of length, time rate and mass will always be extremely small. In the case of Mercury, which is the closest planet to the Sun, these changes will be as small as about one part per billion. Consequently we will regularly simplify the calculations by using only the first order. This will be an excellent approximation. The derivative of the function will then become equal to the finite difference as used in chapter one. This does not change the fundamental understanding of the phenomenon as we will see below.

We have seen in equation 4.1 that the absolute reference meter is a constant number of times (Bo s) the Bohr radius in outer space (a0 s). However, the Bohr radius does not change solely with the gravitational potential. It also changes with velocity. We define the absolute outer space meter as being a meter in outer space with zero velocity. From equation 1.22, the relationship giving the Bohr radius a when there is no change of velocity is (using outer space units):

A« ao.s. - aM

§Ah

4.4

a

a,.

which gives:

^o.s. V

c J

4.5

where mgAh is the change of potential energy (Pot.) of a mass m in a gravitational field across height Ah. In the case of a central force, Newton's laws say that the gravitational potential (Pot.) of a body decreases when the distance (R) from the central body increases. The gravitational potential of a body of mass M(M) (in the case of Mercury) at a distance RM from the Sun of mass M(£) with respect to outer space is:

pot = GM(M)M(I)=M{M)gAh[os]

where G is the Cavendish gravitational constant and g is the gravitational acceleration where the mass is located (here in the solar gravitational field).

74 CHAPTER FOUR. FUNDAMENTAL MECHANISM RESPONSIBLE FOR THE ADVANCE...

In previous chapters, we have used the brackets [rest] and [mov] to indicate the units. From now on, depending on whether we refer to the units of length, mass, clock rate, etc., located in outer space (free from a gravitational potential) or units in the gravitational potential of Mercury, we will use the indices [o.s.] or [M]. The units will always be "translated" in absolute units (e.g. Mercury second = 1.01 absolute seconds). Using equations 4.1, 4.3, 4.5 and 4.6, we find that the length of the Mercury meter (meterM) compared with the absolute reference meter (meter0 s) is:

GM(S)

meter0 s = meterM 1 +

4.7

c2R m 7

We recall that the length of the meter (meter0 s) in outer space is the absolute standard reference. However, we know that when an observer is located on a different frame to measure a given length, he finds a different answer because his unit of comparison (his local meter) is different.

It is useless here to specify the units of GM(S)/c‘Rm. Logically, they should be coherent i.e. either [M] or [o.s.]. Physically, it makes no difference whether the units of G, M(£) or R are the same or not since the error brought in this way is of the order of 1(F9 on GM(S)/c2Rm which is itself of the order of 10‘9 with respect to the meter.

4.3 - The Absolute Reference Second.
An equivalent transformation must be taken into account when time is defined. We can evaluate time on different frames using a local cesium clock. However, one must recall that the rate of such a clock (or of any other clock) changes with the electron mass and therefore with the potential and kinetic energies where the clock is located. Therefore a correction must be made if we want to know the absolute time.
For the case of zero gravitational potential, we now define an absolute time interval called the absolute reference second just as in section 3.5.1 where the second was defined for the case of zero velocity. During one absolute second, a cesium clock makes N(S) (where the index (S) refers to the definition of a second)

CHAPTER FOUR. Fundamental Mechanism Responsible for the Advance... 75

oscillations that are counted from the number of cycles of electromagnetic radiation emitted. That cesium clock must be located outside the gravitational potential of the Sun and have zero velocity. By definition, that absolute time interval will be called the "outer space second". We have:

absolute ref. second = N(S) Oscillations (cesium clocks s). 4.8

During one absolute second, a cesium clock in outer space emitting N(S) cycles shows a difference of clock displays labeled ACDos(S). We must emphasize that ACDos(S) does not correspond to any value of ACD, it corresponds only to the number of counts on the outer space clock leading to the absolute second. This is shown by (S) following the ACD. Consequently, ACD0 s (S) representing the absolute reference second must not be confused with a simple value of ACDframe (without (S)) which can be any number of seconds. We have:

1 abs. sec. = ACD0S (S) = N(S) Oscillations(cesium clocks ). 4.9

When an observer on Mercury observes that his cesium clock has emitted the same number N(S) of cycles, the absolute time interval elapsed is not the absolute second since the Mercury clock is slower. That time interval is called the Mercury second. We have:

1 Mercury sec. = ACDM(S) = N(S) Oscillations(cesium clockM).4.10

Therefore we define one "local second" as the time elapsed when the numerical value shown on a local frame is equal to ACDframe(S). Of course, the Mercury second represented by ACDM(S) lasts longer than the outer space second represented by ACDos(S) because even if the differences of clock displays ACDo s (S) and ACDm(S) are equal, the Mercury clock is slower. Consequently, during one local second, we have for the outer space clock the same ACD than for the Mercury clock:

1 local second = ACDframe(S).

4.11

Since the principle of mass-energy conservation and Bohr equation teach us by how much the rates of two clocks located in outer space and on Mercury differ, an observer on Mercury can calculate the absolute time using his Mercury clock and making

76 CHAPTER FOUR. Fundamental Mechanism Responsible for the Advance...

suitable corrections due to the gravitational potential at Mercury location (we will consider the velocity of Mercury later).

Let us consider that a clock in outer space records a difference of clock displays equal to the number ACD0S. The corresponding absolute time interval elapsed is called Axos [o.s.]. That absolute time interval can be measured on different locations like Mercury or outer space. For a phenomenon taking place in outer space, a time interval can be written:

At0Jo.s.] = ACD0S (o.s.)ACDos(S)

4.12

where Axo s [o.s.] is the absolute time interval, ACD0S (o.s.) is the number of seconds shown by the outer space clock and ACD0 s (S) is the absolute unit of time in outer space given by the o.s. clock.

In equation 4.12, the symbol [o.s.] after Axo s is due to the units of time ACDos(S). The parentheses in ACDo s (o.s.) indicate the units used for the measurement. The subscript o.s. of Ax0 s [o.s.] and ACD0 s (o.s.) refers to the location where the phenomenon takes place (this is different from what we did in chapter three). When an outer space phenomenon is observed using a Mercury clock, the absolute time interval Ax0 s [M] measured on a clock on Mercury is given by the relationship:

Ax0,.[M] = ACD0 s (M)ACDm(S)

4.13

where ACD0S(M) is the number of Mercury seconds and ACDm(S) is the unit of time of the clock located on Mercury, as described in equation 4.10.

Of course, a Mercury second is not equal to one real outer space second. The absolute second is defined in outer space. Therefore a Mercury second is not a real time interval. It corresponds to a difference of clock displays which can be described as an apparent time on Mercury.

If a phenomenon taking place in outer space is measured using a clock located in outer space, its duration will be represented by the absolute time interval Axos [o.s.] (equation 4.12). If this same phenomenon is measured using the Mercury clock, the same absolute time interval will be represented by Ax0S [M] (equation 4.13). Of course, one single phenomenon does not last a longer

CHAPTER FOUR. FUNDAMENTAL MECHANISM RESPONSIBLE FOR THE ADVANCE... 77

absolute time because it is observed from a different location using a different clock. The real absolute duration is the same in any frame. This gives:

At0.s.[o.s.] = Ax0S[M], Using equations 4.12 and 4.13 in 4.14, we find:

4.14

ACD0S (o.s.)ACDos(S) - ACDo s (M)ACDm(S). 4.15

4.3.1 - Example.
In order to clarify this description, let us give a numerical example. Let us assume that an atomic clock located in outer space has emitted 20 times N(S) cycles of E-M radiation. After N(S) cycles, one more absolute second ACD0 s (S) has elapsed and this is repeated ACDos(o.s.) times (with ACD0s(o.s.) = 20). Consequently, the corresponding time interval Ax0 s [o.s.] elapsed is 20 absolute (or outer space) seconds, as given in equation 4.12. That same clock is moved to a stationary location (for example Mercury) near a very massive star so that the relativistic electron mass decreases by 1.0% due to the change of gravitational potential. Quantum mechanics shows that the atomic clock will then run at a rate which is 1.0% slower (as explained in chapter one). Consequently, since the atomic clock on that planet is slower than when it was in outer space, it will take a longer absolute time to make the same number N(S) of oscillations. Since the Mercury second is defined (in equation 4.10) as the time required for the clock on Mercury to emit N(S) cycles, it is longer than the outer space second. This gives:

1 Mercury second =1.01 Absolute second.

4.16

Consequently, during the time interval in which the outer space clock will record an absolute time interval Ato s [o.s.] equal to 20 outer space seconds (ACD0 S (o.s.)), the Mercury clock will record a smaller ACD0S(M) because it runs at a slower rate. The ACD0S(M) recorded on Mercury will be 1.0% smaller:

ACD0S (M) giving the numerical value:

ACP0S (o.s.) 1.01

4.17

78 CHAPTER FOUR. FUNDAMENTAL MECHANISM RESPONSIBLE FOR THE ADVANCE...

ACD0S(M) = —= 19.80198.

4.18

Therefore, in agreement with equation 4.14, since the Mercury second lasts longer, as seen in equation 4.16, the total absolute time elapsed on Mercury (Ato s [M]) is the same as the total absolute time in outer space. We find in equation 4.12:
At0 s [o.s.] = 20x 1 absolute second = 20 absolute seconds. 4.19
From equations 4.13, 4.16 and 4.18 we have: At0S[M] = 19.80198x(1.01 abs. seconds) = 20 abs. seconds. 4.20
Therefore, At is a real absolute time interval in all frames.

4.3.2 - Relative Clock Displays between Frames.
We have seen that the clock used in each frame simply counts the number of cycles emitted by the local atomic clock. In all frames, the local second is equal to the count of N(S) cycles on the local clock. During one absolute time interval, the number of cycles is then proportional to the absolute clock rate which is its absolute frequency as given by equation 1.22 (when v = 0). Therefore, during one absolute time interval, the ratio of the differences of clock displays between frames is directly proportional to the ratio of the natural frequency of each clock. This gives:
ACP0S (o.s.) _ vos
ACD0S (M) vM '
Equation 4.21 gives the relative frequencies of clocks located in different frames. Obviously, it does not matter whether the phenomenon measured is in outer space or on Mercury, as long as both clocks measure the same phenomenon. This means that the subscripts of the left hand side of 4.21 could both be M instead of o.s.. If there is a difference of kinetic energy between the frames, equation 3.9 must be applied. Any difference of clock rate is caused by the difference of gravitational potential and/or kinetic energy between an outer space location and the orbit of Mercury. In the case of pure potential energy, using equations 1.22 and 4.6, the relative clock rate is given by the relationship:

CHAPTER FOUR. Fundamental Mechanism Responsible for the Advance... 79

Av _ vo.s. - vM _ gAh ^ M(S)G

v

vos

c2 “ c2Rm

4.22

which gives:

Jx V,.,

GM(S)V

VM

V

C R-M '

4.23

Using equation 4.21 with equation 4.23, we see that during the same absolute time interval, the relative difference of clock displays is:

ACDm(o.s.) _ ACD0S (o.s.) _ vos ACDm(M) ACDos(M) vM

GM(S) . 4.24

Let us note that these equations do not take into account a second order that might exist when the particle moves down in the gravitational potential. Since that second order effect is quite negligible in the first chapters of this book, we will consider it only if it becomes significant.

4.4 - The Absolute Reference Kilogram.

The absolute unit of mass is also defined in outer space. We

have seen in chapter one that one absolute kilogram (kg0 s) in outer

space contains a different amount of mass after it is carried to

Mercury. When we carry a mass of one kilogram (kgos) from

outer space to Mercury location (at rest), the amount of mass

decreases (because it gives up energy during the transfer).

However, the observer on Mercury will still call it one Mercury

kilogram (kgM) since the number of atoms has not changed. In

fact, nothing appears to change for an observer moving with the

kilogram and observing a physical phenomenon on Mercury. The

relationship between two kilograms located in different potentials

is given in equation 1.5. Using equations 1.5 and 4.6, we find:

f GM(S)N
1 kgM = kgo.s. V c R-m 2

4.25

Equation 4.25 gives the mass of the outer space kilogram with respect to the Mercury kilogram.

80 CHAPTER FOUR. FUNDAMENTAL MECHANISM RESPONSIBLE FOR THE ADVANCE...
4.5 - Space and Time Corollaries within the ActionReaction Principle.
Let us discuss what happens inside a frame located at the position where Mercury interacts with the Sun's gravitational field. What is the behavior of Newton's laws at that location?
We believe in the principle of causality. The cause is the reason for the action. Newton applied this principle and stated that an action is always accompanied by a reaction. However, even if this has not been stated specifically, it becomes obvious that there are two corollaries to that principle. The first corollary is that both the action and the reaction take place at exactly the same location where the interaction takes place. The second corollary is that both the action and the reaction take place at exactly the same time the interaction takes place. The principle of causality implies that it is illogical and indefensible to believe that the cause of a phenomenon does not take place at the same location and at the same time that the effect does.
Let us apply those corollaries to relativity. When a mass moves in a gravitational field, its trajectory is modified by the action of the gravitational field. The interaction between a mass and a gravitational field takes place at the location of the mass and at the moment the mass is interacting with the field. Consequently, the relevant parameters during the interaction are the amount of mass and the intensity of the gravitational field at the location of the interaction. It would be absurd to calculate an interaction using quantities that exist somewhere else than where the interaction takes place. When we study the behavior of Mercury interacting with the solar gravitational potential, we must logically use the physical quantities existing where Mercury is located. This means that when we calculate the behavior of planet Mercury, we must use the units of length, clock rate and mass existing at Mercury location. This is the only logical way to be compatible with the principle of causality and with its natural corollaries leading to the principle of action-reaction. It would not make sense for the mass of Mercury involved in the interaction with the solar gravitational field to be the mass it has in outer space rather than its real mass where it is located at the moment it is interacting near the Sun.

CHAPTERFOUR. Fundamental Mechanism Responsible for the Advance... 81
Therefore the amount of mass, length and clock rate that must be used in the equations are the ones that appear at Mercury location, since they are the only relevant parameters logically compatible with the physics taking place on Mercury. At Mercury location, there is no other physics than the one using the local mass, length and clock rate. Logically, it must be so everywhere within any frame in the universe. This point is extremely important and is fundamental in the calculations below because it is the basic phenomenon that explains the advance of the Mercury perihelion around the Sun.
4.6 - Fundamental Mechanism Taking Place in Planetary Orbits.
In classical mechanics, it is demonstrated that planets revolve around the Sun in a circular or elliptical orbit. The complete period of an orbit can be defined as the time taken to complete a full translation of 2n radians around the Sun or as the time interval taken by the planet to complete its ellipse between the passages of a pair of perihelions. It is usually considered that these two definitions of a period of an orbit are identical. However, if the ellipse is precessing, the angle spanned between the two passages of a pair of perihelion is larger than for a non precessing ellipse i.e. larger than 2n radians. This means that the full translation of 2n radians is completed before the ellipse reaches the next perihelion. Therefore we expect the period of that precessing ellipse to be larger.
One of the fundamental phenomena implied in such an orbital motion is the gravitational potential decreasing as the inverse of the distance from the Sun where the planet is orbiting. When the orbit is circular, it is difficult to determine at what instant one full orbit is completed other than measuring a translation of 2n radians with respect to masses seen in outer space. However, in an elliptical orbit (as in the case of Mercury around the Sun), the direction of the major axis can be easily located in space from the instant Mercury is at its perihelion, i.e. its closest distance from the Sun.

82 CHAPTER FOUR. FUNDAMENTAL MECHANISM RESPONSIBLE FOR THE Advance...

4.6.1 - Significance of Units in an Equation.
In Galilean mechanics, when the units are identical in all frames, the pure number that multiplies the unit is undistinguishable from the quantity that includes the unit. For example, when someone reports that a rod is ten meters long, we can assume that either he has in mind that the rod is ten times the length of the standard meter (in which ten is a pure number separated from the unit of length), or he means a single global quantity with unit, corresponding to one single quantity ten times longer than the unit meter. Of course, the difference brings no consequence at all when we always use the same standard meter. However, the correct interpretation must be understood and specified here because the size of the reference meter (and all other units) changes from frame to frame.
If "a" represents the semi-major axis of the elliptical orbit of Mercury, we have to find whether "a" represents a pure number (to which a unit is added and considered separately) or a single global quantity (with units included). This can be answered if we study the fundamental role of a mathematical equation. In mathematics, we learn that an equation is a fundamental relationship between numerical quantities. The same mathematical equation can relate numbers (or concepts) having different units. This can be illustrated in the following way.
If an apple costs 50 cents, how many apples (N) will we buy with $10.00? We use the following equation:

4.26

With a — $10.00, and b = $0.50 each, we find

N = 20 apples.

4.27

Now, if we also find that an orange costs 50 cents, how many oranges will we have for $10.00? Using again equation 4.26 with a = $10.00 and b = $0.50 each, we find:

N = 20 oranges.

4.28

CHAPTER FOUR. Fundamental Mechanism Responsible for the Advance... 83

We also want to buy peas. They cost 1 cent each. How many peas do we get for $10.00? Using again equation 4.26, we find that the number of peas is:

N= 1000 peas.

4.29

Equations 4.27, 4.28 and 4.29 illustrate that the mathematical parameter N does not represent apples, oranges or peas. It represents only the numerical value of the unit. The unit must be specified separately. One must know that the units also follow a separate mathematical relationships. This is called a dimensional analysis which requires an analysis separate from the numerical analysis.

Therefore, "a" represents the number of units of length. The same remark must be applied to all physical quantities that are pure numbers obtained from a previous definition of other standard units. Furthermore, in order to be compatible with the principle of causality given above, the units of length, mass and clock rate must necessarily be the ones existing on Mercury where the phenomenon takes place. We will see below how this description leads to a perfect coherence.

In the solar system, the orbit of Mercury is very elongated and is an excellent example to study Kepler's laws. However, since there are several other planets moving around the Sun, there are other classical corrections due to the interactions between these other planets that need to be taken into account. Extensive classical calculations show that the interaction of the other planets of the solar system also produces an important advance of the perihelion of Mercury. After accurate calculations, data show that the advance of the perihelion of Mercury is larger than the value predicted by classical mechanics. The advance of the perihelion is observed to be 43 arcsec per century larger than expected from all classical interactions by all planets.

In order to solve this problem, we have to examine in more detail the conditions in which the equations must be applied. As we will see in chapter five, the number of seconds giving the period P is a function of the parameters a, G, M(S) and M(M). However, due to mass-energy conservation we have seen that the units of length,

84 CHAPTER FOUR. FUNDAMENTAL MECHANISM RESPONSIBLE FOR THE ADVANCE...

time and mass are different at Mercury distance from the Sun than in outer space. In section 4.5, we have also seen that the action of the gravitational potential on Mercury must be calculated using the number of units of mass (and all other parameters) that Mercury has at that location.

4.7 - Transformations of Units. 4.7.1 - aM(o.s.) versus aM(M).
When we measure the number of meters that constitute a given length, we find that this number depends on the length of the unit used in conjunction with it. We call aM(o.s.), the number of outer space meters that represents the length of the semi-major axis of the orbit of Mercury when we use outer space meters. The absolute physical length LM[o.s.] being measured using outer space meters is then:

Lm[o.s.] = aM(o.s.)meter0 s.

4.30

The value of the absolute length LM[o.s.] of the semi-major axis of the orbit of Mercury corresponds to measuring the number aM(o.s.) of meters in the orbit times the outer space meter (meteros). We now have to determine the number aM(M) of Mercury meters (meterM) found in conjunction with Mercury units. aM(M) represents the corresponding number of Mercury meters to measure the same length when we use Mercury meters. We find that the absolute physical length Lm[M] of the semi-major axis, is given by:

Lm[M] = aM(M)meterM.

4.31

Since a physical length does not change because we use a different reference meter to measure it, we must understand that the absolute physical length of the semi-major axis is the same whether it is measured using outer space or Mercury units. Therefore, the absolute length LM[frame] of the semi-major axis of the orbit of Mercury is the same independently of the units used to measure it. Therefore, equations 4.30 and 4.31 are identical:

Lm[M] = Lm[o.s.] = aM(o.s.)meteros = aM(M)meterM. 4.32

CHAPTER FOUR. Fundamental Mechanism Responsible for the Advance... 85

Equation 4.32 gives us the relationship between the number aM(o.s.) of outer space meters and the number aM(M) of Mercury meters to measure the same length. This gives:

aM(o.s.) = aM(M) meter,M meter„

4.33

Combining equations 4.7 and 4.33 gives:

GM(S) aM (o.s.) = aM (M) 1 +
c Rm )

4.34

Equation 4.34 shows that the number aM(M) of Mercury meters required to equal the semi-major axis of Mercury is smaller than the number aM(o.s.) of outer space meters since the outer space meter is shorter. Therefore the outer space observer will record a larger number aM(o.s.) of meters than the Mercury observer even if both observers are measuring the very same semi-major axis.

4.7.2 - M(£)(o.s.) and M(M)m(°-s>) versus M(S)(M) and M(M)m(M).
The symbols (&) and (M) represent respectively the Sun and Mercury. M(£)(o.s.) and M(M)m(°-s-) represent the numbers of absolute outer space kilograms (kgos) for the Sun and Mercury respectively. The subscript M of M(M)m(°-s-) indicates that the planet is at Mercury location. The numbers of Mercury units that give the same masses are represented by M(£)(M) and M(M)m(M). The absolute solar mass p(£)[o.s.] using outer space units is:

p(S)[o.s.] = M(S)(o.s.)kg04.35

Using Mercury units, the same absolute solar mass is given by:

p(£)[M] = M(S)(M)kgM.

4.36

Since the solar mass does not change because we measure it using Mercury units instead of outer space units, we have:

p(S)[o.s.] = p(£)[M].

4.37

Similarly, the mass of Mercury measured with outer space units is:

p(M)M[°-s-j = M(M)M(o.s.)kg0.s..

4.38

86 CHAPTER FOUR. FUNDAMENTAL MECHANISM RESPONSIBLE FOR THE ADVANCE...

When the measurement is done with Mercury units, the same mass is given by:

|.i(M)m[M] = M(M)M(M)kgM.

4.39

Since it is the same absolute mass of Mercury described using different units, we have:

|t(M)m[o.s.] =

4.40

Due to mass-energy conservation, the amount of mass contained in one local Mercury kilogram is different from the one in one outer space kilogram. From equations 4.35, 4.36 and 4.37 we have:

M(S)(o.s.) = kgM M(S)(M) kg0,'

441

The left hand side of equation 4.41 gives the ratio between the number of outer space kilograms and the number of Mercury kilograms needed to measure the same solar mass. From equation 4.25, we get:

kgo.s.
kg M

Vl
GM(S) c2R m 7

4.42

Combining equations 4.41 and 4.42 gives:

M(S)(o.s.) = M(S)(M)

GM(S) c Rm J

4.43

Equation 4.43 shows that the number of kilograms M(£)(o.s.) found in the measurement of the solar mass is smaller when measured in conjunction with the outer space kilogram than when measured in conjunction with the Mercury kilogram. Combining equations 4.38, 4.39 and 4.40 with 4.42, we get for the case of the mass of Mercury:

M(M)m(o.s.)=M(M)m(M)

GM(S)" c2Rm >

4.44

Consequently, the number M(M)m of kilograms giving the mass of Mercury is smaller using outer space kilograms than using Mercury kilograms.

CHAPTER FOUR. FUNDAMENTAL MECHANISM RESPONSIBLE FOR THE ADVANCE... 87

4.7.3 - PM(o.s.) versus Pm(M).
In equations 4.12 and 4.13, we have calculated absolute time intervals At as measured from outer space location (Axo s [o.s.]) and Mercury location (Axo s [M]). Let us consider now that the time interval Ax is the period of translation of Mercury to complete an ellipse around the Sun. The number of seconds PM(o.s.) giving the period of Mercury when measured with an outer space clock is given by the relationship:

Atm[o.s.] = PM(o.s.) ACDo s (S)

4.45

and the period Pm(M) measured on Mercury using a Mercury clock (with Mercury units) refers to the relationship:

Atm[M] = Pm(M)ACDm(S).

4.46

The time intervals Atm[o.s.] and Atm[M] in equations 4.45 and 4.46 represent the absolute time interval for the period P of translation of Mercury around the Sun. An absolute time interval is not different because it is measured with a Mercury clock instead of an outer space clock:

Axm[o.s.] = Atm[M] = Pm(M) ACDm(S) = PM(o.s.) ACDos(S).4.47

We have seen in equation 4.24 the ratio of the numbers ACDm(o.s.) and ACDm(M) between two frames in different gravitational potentials. We see that the numbers PM(o.s.) and Pm(M) displayed by the clocks correspond to ACDM(o.s.) and ACDm(M) during one period of translation. Therefore,

ACDm(o.s.) PM(o-s.) ACDm(M) Pm(M) '

4.48

Combining equation 4.48 with 4.24 gives:

PM(o.s.) ACDm(o.s.) Pm(M) ACDm(M)

GM(S) 1-
c2Rm )

4.49

Equation 4.49 shows that even if the absolute time interval Ax for the period is the same in both frames, the differences of clock displays are different because the clocks run at different rates.

88 CHAPTER FOUR. Fundamental Mechanism Responsible FOR the Advance...

4.7.4 - G(o.s.) versus G(M).
Since lengths, clock rates and masses are not the same in different frames, we see now that the gravitational constant G is different when measured using Mercury units. The number of outer space units of the gravitational constant is called G(o.s.) and the number of Mercury units of the same gravitational constant is called G(M). The fundamental units corresponding to the gravitational constant G are called respectively Uos and UM. The total gravitational constant G is called J[o.s.] when measured from outer space and J[M] when measured from Mercury orbit. Therefore we have:

J[o.s.] = G(o.s.)U0 s and

4.50

J[M]=G(M)Um.

4.51

Since the absolute gravitational constant does not change because we measure it from a different location, we have:

J[o.s.]=J[M].

4.52

The relative number of units between G(o.s.) and G(M) is found using a dimensional analysis. The units of G can be obtained from Newton's well known gravitational law:

GMm

4.53

where the force F is in newtons, M and m are in kilograms and the radius R is in meters. From equation 4.53 and recalling that the units of G(o.s.) are called Uos, we find:

newtonosmeter0z&

4.54

From the relationship

F = mot

4.55

where a is the acceleration, we find that the units of F are:

newton0..

kgo.s.meter0s
SeCo.s.

Combining 4.54 with 4.56 we get:

4.56

CHAPTER FOUR. Fundamental Mechanism Responsible for the Advance... 89

u o.s.

kgasmeter03s
kgo.s. seCo.s. '

From the definition of velocity, the units of v are:

meter. secn

Equation 4.58 in 4.57 gives:

4.57 4.58

meter0 &Vq s. ' r\ c
kgo.S

4.59

We have seen in sections 3.5.3 and 3.6 that a velocity is represented by the same number within any frame. This means that the number representing a velocity is the same within any frame

when it is measured using any coherent system of local units. Since a velocity is the quotient between a length and a time interval, this quotient stays constant even when switching between frames because the same correction is made on both lengths and clock displays. Consequently, we have:

v0.s. = vM.

Equations 4.7, 4.42 and 4.60 in equation 4.59 give:

GM(S) vi

meter,M 1 +

'M

c2Rm )

Uo.s. =

\-i
GM(S) kg M 1-
c2Rm )

The first order expansion of equation 4.61 gives:

4.60 4.61

meterMv^ U„, =
kg M

GM(S)
) c2Rm

By analogy with 4.59 for UM, we have:

meterMMVvM UM = kg M

Equation 4.63 in 4.62 gives:

Ur u M

A2 GM(S) c2R M 7

4.62 4.63 4.64

90 CHAPTER FOUR. FUNDAMENTAL MECHANISM RESPONSIBLE FOR THE ADVANCE...

Equations 4.50, 4.51, 4.52 and 4.64 give the relationship between the number of units of G:

GM(S)V2 G(o.a) = G(M) 1-
c2Rm >

4.65

Equation 4.65 shows that the gravitational constant G is represented by different numbers when measured with the units existing on Mercury and in outer space.

4.7.5 - F(o.s.) versus F(M).
From equation 4.56 we have:

newton

= kgo.s.metero.s. sec.

Using equations 4.7, 4.15, 4.24 and 4.25, we find:

4.66

kg>M
V
neWt0no.s. =

\ GM(S)A

meter,M

c Rm )

V

c R■M 7

\2

GM(S)

sec M 1l

c

Rm

J

4.67

To the first order, this is equal to:

and:

kgM meter.

newtono.s. =

secM

\2 GM(S) 1 +

^

c Rm )

4.68

\2 GM(S) newtom 0 = newton M 1 +
c Rm )

4.69

Consequently, the relationship between the number of Mercury newtons and the number of outer space newtons is given by:

_

/ GM(S)V2

F(o. s.)= F(M)| 1+ 2

\

cR M 7

4.70

4.8 - Symbols and Variables.
aframe[°-s-] length of the local Bohr radius in absolute units

CHAPTER FOUR. Fundamental Mechanism Responsible for the Advance... 91

&m(M)

number of Mercury meters for the semi-major axis of

Mercury

aM(o.s.)

number of outer space meters for the semi-major axis of Mercury

ACDm(M) ACD for the period of Mercury measured by a

Mercury clock

ACDm(o.s.) ACD for the period of Mercury measured by an outer

space clock

ACDm(S) apparent second on Mercury

ACDos (M) ACD in outer space measured by a Mercury clock

ACD0S(o.s.) ACD in outer space measured by an outer space clock

ACD0,.(S) absolute second in outer space

Atm[M]

period of Mercury in Mercury units

Atm[o.s.] period of Mercury in outer space units

At0,.[M]

time interval in outer space in Mercury units

At0.s.[o.s.] time interval in outer space in outer space units

G(M)

number of Mercury units for the gravitational

constant

G(o.s.)

number of outer space units for the gravitational

constant

J[M]

gravitational constant in Mercury units

J[o.s.]

gravitational constant in outer space units

frame
Lm[M]

mass of the local kilogram in absolute units length of the semi-major axis of the orbit of Mercury

in Mercury units

Lm[o.s.]

length of the semi-major axis of the orbit of Mercury

in outer space units

meterframe length of the local meter in absolute units

M(M)m(M) number of Mercury units for the mass of Mercury at

Mercury location

H(M)m[M] mass of Mercury in Mercury units at Mercury

location

M(M)m(o-s.) number of outer space units for the mass of Mercury

at Mercury location

|t(M)m[o-s.] mass of Mercury in outer space units at Mercury

location

M(S)(M) number of Mercury units for the mass of the Sun

M(S)(o.s.) number of outer space units for the mass of the Sun

92 CHAPTER FOUR. FUNDAMENTAL MECHANISM RESPONSIBLE FOR THE ADVANCE...

C ?3

|a(S)[M] p(£)[o.s.] N(S)
Pm(M)
Pm(o.s.)
M frame

mass of the Sun in Mercury units mass of the Sun in outer space units number of oscillations of an atomic clock for one local second ACD for the period of Mercury measured by a Mercury clock ACD for the period of Mercury measured by an outer space clock distance between Mercury and the Sun unit of the gravitational constant in the local frame

Chapter Five Calculation of the Advance of the Perihelion of Mercury.
5.1 - Mathematical Transformation of Units between Frames.
In this chapter we will deal with two kinds of transformations. The first kind is a mathematical transformation of units which brings no physical change to the quantities being described. In such a transformation, there is no physics, just mathematics. For example, let us suppose that we measure a rod on Mercury and find that it is 100 times longer than the local Mercury meter. Then we say that the length of the rod is 100 Mercury meters. However, if we know that on Mercury, the local meter is 1% longer than the local reference meter in outer space, we know that the same rod is actually equal to 101 times the outer space reference meter. These two descriptions by units of different frames are perfectly identical. The rod has not changed.
The observer on Mercury can also use his clock to measure a time interval. If the Mercury observer measures 100 units on his clock (i.e. 100 Mercury seconds), knowing that clocks on Mercury run at a rate which is 1% slower than clocks in outer space, we can calculate that during that absolute time interval the difference of clock displays on a clock in outer space will be 101 outer space units. No physics is involved in that transformation, only mathematics. The same physical phenomenon is described using different units.
Other units must also be transformed. For example, the absolute mass of the Sun does not change because we observe it from Mercury location near the Sun. However, measuring the same solar mass using the smaller Mercury unit of mass will lead to a larger number of Mercury units. Similarly, the physical amplitude of the absolute gravitational constant G does not change because the phenomenon takes place near the Sun. We have seen in chapter four that the absolute constant G is represented by different

94 CHAPTER FIVE. Calculation of the Advance of the Perihelion of Mercury.
numbers of Mercury and outer space units. Again, no physics is involved.
5.1.1 - Consequence of a Simple Change of Units.
Let us suppose that using Newton's relationships, we want to calculate the period of Mercury using Mercury units. We must then compare this answer with the one obtained with the same relationships using outer space units. If we do so, we find that the numbers of units found for the period are different. However, when we take into account that the Mercury clock runs at a slower rate, we see that the absolute times obtained from either frame are the same.
In the next section we will see that in order to be compatible with the principle of mass-energy conservation, one must add another kind of transformations which are physical transformations. Contrary to the identical consequences resulting from the mathematical transformation explained above, different absolute results are found when Newton's laws are applied with the proper values belonging to different frames.
5.2 - Physical Transformations Due to Mass-Energy Conservation.
The second kind of transformations consists of real physical changes. We have seen in chapters one and three that when an object in outer space is moved to Mercury location, its absolute mass changes because of the change of gravitational potential and kinetic energies. (In the case of gravitational energy, the difference of mass is transformed into work). The object that remains at Mercury location is physically different from the object that existed in outer space because the dimensions of its atoms, their mass and clock rate have changed. This physical change of mass is quite different from the mathematical change of units mentioned above.
Here is an example. An observer on Mercury measures that a mass on his frame is 100 times larger that the unit of mass on Mercury. Another observer in outer space measures the mass of the same object after it has been carried out to outer space. In that new frame, the outer space observer finds the same number (100)

CHAPTER FIVE. CALCULATION OF THE ADVANCE OF THE PERIHELION OF MERCURY. 95
of new units of mass. Both observers measure 100 local kilograms. However, the absolute mass of this object has changed when moved from Mercury location to outer space. The Mercury kilogram is not equal to the outer space kilogram. To realize this, we need to know the mass at Mercury location using outer space units. Applying the principle of mass-energy conservation, we find that using the same outer space units, the mass of the object is reduced to only 99 outer space kilograms when brought to Mercury location (since the Mercury kilogram is 1% lighter than the outer space kilogram). This is a real physical change. It is not a simple mathematical transformation of units like the one explained in section 5.1.
We will see in section 5.3 that these physical changes lead to results that are physically different when calculated using proper values in different frames. Using Newton's classical mechanics, we will find that the results obtained using the proper parameters in one frame are not coherent with the results obtained using parameters proper to another frame.
In order to clarify this description, in this chapter we will use the expression transformation of units to designate only a pure mathematical transformation of units. When a physical change is involved as a consequence of mass-energy conservation, we will speak of a transformation of parameters.
We consider that the interactions between physical elements (like fields, masses, lengths and clock rates) existing on Mercury, using Mercury parameters, must be the same as the ones that we calculate in outer space using outer space parameters. This means that the mathematical relationships so well-known in physics are the only ones that are valid but it is required that on Mercury we use the physical quantities (mass, length and clock rate) existing on Mercury while in outer space, we use the physical quantities (which are different) existing in outer space. In other words, we must always use proper values. It is totally illogical to use outer space physical parameters at Mercury location. On Mercury, we must necessarily use physical parameters that exist on Mercury.

96 CHAPTER FIVE. CALCULATION OF THE ADVANCE OF THE PERIHELION OF MERCURY.
5.3 - Incoherence between Outer Space and Mercury Predictions Using Newton's Physics.
In this book, we use Newton's equations which are always perfectly valid in all frames. However, there is a difference between Newton's equations and Newton's physics. Newton's physics is different from the physics described in this book because it is not compatible with the principle of mass-energy conservation. In Newton's physics, there is no place for changes of mass, length and clock rate. According to that physics, the mass of an object in outer space does not change if it is transported to Mercury location or to anywhere in the universe.
Let us suppose a Newtonian observer wants to measure the period of Mercury. He wishes to know its mass. To do this, he imagines the following thought experiment. He takes Mercury out of its orbit to outer space and puts the planet on a balance to measure its mass. Then he puts Mercury back on its orbit. Being a Newtonian observer using Newton's physics, the mass he will use in his calculations of Mercury's period will be the mass he just measured in outer space. However, we know this mass is wrong because of mass-energy conservation. We also know that other parameters (like length and clock rate) at Mercury location are modified due to the change of mass. Therefore this observer's Newtonian calculation of the orbit of Mercury will be wrong even when he uses the correct equations.
We will see that when the orbit of a planet moving around the Sun is calculated, using outer space physical parameters, we find a perfect ellipse. However, when we use the proper parameters existing on Mercury, we find a different orbit which is a precessing ellipse. This explains the advance of the perihelion of Mercury. When neglecting the changes of mass, length and clock rate on Mercury with respect to outer space, we find an erroneous prediction because we use outer space physical parameters instead of proper parameters.

CHAPTER FIVE. CALCULATION OF THE ADVANCE OF THE PERIHELION OF MERCURY. 97

5.4 - Incoherence of the Gravitational Force Using Newton's Physics.
Let us give an example that shows that the calculated force of gravity is different depending on what the physical parameters are used (outer space or Mercury). For the Newtonian observer, the gravitational force is:

M(M)0S (o.s.)

Fg (o.s.) = G(o.s.)M(S)(o.s.)

5.1

Rm(oS-)

For that observer, whether the subscript of M(M) is o.s. or M makes no difference. We write o.s. because this observer uses Newton’s physics which always assumes a constant mass. The relevant physical parameters at Mercury location are:

M(M)m(M)

Fg (M) = G(M)M(S)(M)

5.2

Rm(M)

All physical parameters in equation 5.2 must be Mercury physical parameters because that is where the interaction takes place. We will now compare these two equations. We know that the number of Mercury units to measure the mass of Mercury at Mercury location is the same as the number of outer space units to measure the mass of Mercury in outer space. This gives:

M(M)m(M) = M(M)0,.(o.s.).

5.3

The relationship between the number of units of mass of the Sun in outer space and Mercury units is given by equation 4.43:

GM(S)

M(S)(o.s.) = M(S)(M) 1-

5.4

c2Rm )

The relationship between the numbers of meters to measure the distance of Mercury from the Sun in outer space and Mercury units can be deduced from equation 4.34:

GM(S)"

Rm(o.s.) = Rm(M) 1 +

5.5

c2Rm ,

Finally, the corresponding relationship for the gravitational constant G is given by equation 4.65:

98 CHAPTER FIVE. CALCULATION OF THE ADVANCE OF THE PERIHELION OF MERCURY.

f

gm(S)Y

G(o.s.) = G(M) 1 +

5.6

V c2Rm v

Equations 5.3, 5.4, 5.5 and 5.6 in 5.2 give:

\ | GM(S)|M(M)0,(o.s.) 57 Fg (M) = G(o.s.)M(S)(o.s.)
c-Rm ) Rm(o.s.)
In order to compare the gravitational force calculated using Mercury units, with the force calculated using outer space units, let us transform the number of units of force FG(M) into the corresponding number of outer space units. From equation 4.70, we have:

gm(S)Y

Fg(M) = Fg(o.s.) 1 +

5.8

C2Rm >

Equation 5.7 with 5.8 gives:

GM(S) Fg (o.s.) = G(o.s.)M(S)(o.s.) 1 +

M(M)os.(0-S-)

c2Rm 7

Rm (o.s.)

We must notice that equation 5.9 does not corresponds to a simple transformation of units. The physical parameters existing on Mercury at Mercury location have been taken into account.

Using the physical parameters existing on Mercury and outer space units, equation 5.9 shows that the absolute gravitational force on Mercury is different from the one calculated using the physical parameters existing in outer space and given in equation 5.1. The two results are not compatible. They predict different orbits. As explained above, the logical choice requires that we choose the equation obtained using the proper physical parameters existing at the location where the interaction of Mercury takes place with the gravitational held. We must reject the calculation obtained using outer space parameters when the experiment is taking place on Mercury. Finally, we now realize that equations 5.1 is the limit of equations 5.9 when RM-»oo.

There is another direct consequence of mass-energy conservation. Contrary to equation 5.1, we see in equation 5.9 that using the physical parameters existing on Mercury, the decrease of